Trigonometric and Periodic Functions · Chapter 5 begins Unit 2: Trigonometric and Periodic...

222222

Chapter 5 Periodic Functions and Right Triangle Problems

Chapter 6 Applications of Trigonometric and Circular Functions

Chapter 7 Trigonometric Function Properties and Identities and Parametric Functions

Chapter 8 Properties of Combined Sinusoids

Chapter 9 Triangle Trigonometry

y

y

x

y 1

x

y

x

Ca

c

b

BA

Trigonometric and Periodic Functions

Unit Overview e central focus of this unit is a study of trigonometric and periodic functions In Chapter 5 students are introduced to the sine and cosine functions the six trigonometric function de nitions inverse trigonometric functions and solving right-triangle problems In this chapter the domains of the trigonometric functions are acute angles measured in degrees Radian measure is introduced in Chapter 6 allowing students to expand the possible domain values of the trigonometric functions to be all real numbers thus generating the corresponding circular functions Used in this way the domain might represent time or distance making circular functions particularly useful for studying real-world applications sinusoidal functions are used as mathematical models to make predictions In Chapter 7 students explore trigonometric properties and identities is chapter covers basic trigonometric properties such as the Pythagorean identities Proving identities helps students learn the properties sharpen their algebraic skills and practice writing algebraic proofs Students also learn to use these properties to solve trigonometric equations In Chapter 8 students extend their study of trigonometric properties to some more complicated properties Graphical investigations of sums and products of sinusoids with unequal periods support an algebraic study of these properties In Chapter 9 students learn to solve problems involving oblique triangles using the law of cosines and the law of sines Students also study the ambiguous case of a triangle and are introduced to vectors

Using This Unit e study of periodic functions follows logically from the rst unit on algebraic exponential and logarithmic functions Unit 2 is central to any precalculus course and students preparing to take calculus will need a solid grasp of the concepts If you want to study trigonometry early in the year this unit can be taught following Chapter 1 without impacting the remainder of the courseFor students who have not studied trigonometry before Chapter 5 provides a solid foundation If students have already mastered this material you may wish to do a few review problems and move on to the next chapter

245

Ice-skater Michelle Kwan rotates through many degrees during a spin Her extended hands come back to the same position at the end of each rotation us the position of her hands is a periodic function of the angle through which she rotates (A periodic function is a function that repeats at regular intervals) In this chapter you will learn about some special periodic functions that can be used to model situations like this

Periodic Functions and Right Triangle ProblemsPeriodic Functions and

555555

y

245

CHAP TE R O B J EC TIV ES

bull Findthefunctionthatcorrespondstothegraphofasinusoidandgraphitonyourgrapher

bull Givenanangleofanymeasuredrawapictureofthatangle

bull Extendthedefinitionsofsineandcosinetoanyangle

bull Beabletofindvaluesofthesixtrigonometricfunctionsapproximatelybycalculatorforanyangleandexactlyforcertainspecialangles

bull Giventwosidesofarighttriangleorasideandanacuteanglefindmeasuresoftheotherside(s)andangles

245A Chapter 5 Interleaf Periodic Functions and Right Triangle Problems

OverviewInthischapterstudentsareintroducedtoperiodicfunctionsastheyanalyzethemotionofaFerriswheelTh eyploty5sinxontheirgraphersandthenusethedilationsandtranslationstomaketheresultingsinusoidfitthereal-worldsituationTh ereafterstudentsareshownhowtoextendthefamiliarcosineandsinefunctionstoanglesgreaterthan180orlessthan0byconsideringanangleasameasureofrotationByfindingvaluesofallsixtrigonometricfunctionsonthecalculatorstudentslearnthereciprocalpropertiesthuspavingthewayfortheformalstudyofpropertiesandidentitiesinChapter7Chapter5concludeswithrighttriangletrigonometrypossiblyafamiliartopicforthemainpurposeofintroducingtheinversetrigonometricfunctionsinameaningfulcontextaswellasprovidingpracticewiththedefinitionsofthesixfunctions

Using This ChapterChapter5beginsUnit2Trigonometric and Periodic FunctionsandprovidesanintroductiontoperiodicfunctionsandtheirgraphsFollowingUnit1thischapterexpandstheideaofafunctiontoincludeperiodicandtrigonometricfunctionsandprovidesasolidfoundationforintroducingstudentstoradiansinChapter6IfyouwishtocovertrigonometryearlierintheschoolyearthischapterflowseasilyfromChapter1Studentswhohavenotstudiedtrigonometryshouldnotskipthischapter

Teaching ResourcesExplorationsExploration5-1aTransformedPeriodicFunctionsExploration5-2ReferenceAnglesExploration5-3aDefinitionsofSineandCosineExploration5-3buv-Graphsanduy-GraphsofSinusoidsExploration5-3cParentSinusoidsExploration5-4ValuesoftheSineandCosineFunctionsExploration5-4aValuesoftheSixTrigonometricFunctionsExploration5-4bDirectMeasurementofFunctionValuesExploration5-5aMeasurementofRightTrianglesExploration5-5bAccurateRightTrianglePracticeExploration5-5cEmpireStateBuildingProblem

Blackline MastersTrigonometricRatiosTableSection5-3

Supplementary ProblemsSections5-4to5-6

Assessment ResourcesTest12Sections5-1to5-4FormsAandBTest13Section5-5FormsAandBTest14Chapter5FormsAandB

Technology ResourcesDynamic Precalculus ExplorationsSineWaveTracer

ActivitiesSketchpadASineWaveTracerSketchpadTrigonometryTracersCASActivity5-2aMeasurementConversionsCASActivity5-4aPythagoreanRelationships

Periodic Functions and Right Triangle ProblemsPeriodic Functions and Right Triangle Problems

C h a p t e r 5

245BChapter 5 Interleaf

Standard Schedule Pacing Guide

Block Schedule Pacing Guide

Day Section Suggested Assignment

1 5-1 IntroductiontoPeriodicFunctions 1ndash4

2 5-2 MeasurementofRotation RAQ1ndashQ10159192125272930

3 5-3 SineandCosineFunctions RAQ1ndashQ101ndash23odd

45-4 ValuesoftheSixTrigonometricFunctions

RAQ1ndashQ101(4)67911ndash14

5 15ndash1921ndash31odd353940ndash46

6 5-5 InverseTrigonometricFunctionsandTriangleProblems RAQ1ndashQ101ndash57(8)9ndash111314(21)24

75-6 ChapterReview

R0ndashR5T1ndashT22

8 EitherC1andC3orProblemSet6-1


15-2 MeasurementofRotation RAQ1ndashQ10159192729

5-3 SineandCosineFunctions RAQ1ndashQ101ndash13odd

25-3 SineandCosineFunctions 15ndash23odd

5-4 ValuesoftheSixTrigonometricFunctions RAQ1ndashQ10167911ndash14

35-4 ValuesoftheSixTrigonometricFunctions 151923273346

5-5 InverseTrigonometricFunctionsandTriangleProblems RAQ1ndashQ101ndash13odd24

4 5-6 ChapterReview R0ndashR5T1ndashT20

55-6 ChapterTest

6-1 SinusoidsAmplitudePeriodandCycles 1ndash10

247Section 5-1 Introduction to Periodic Functions

Exploratory Problem Set 5-1

Introduction to Periodic FunctionsAs you ride a Ferris wheel your distance from the ground depends on the number of degrees the wheel has rotated (Figure 5-1a) Suppose you start measuring the number of degrees when the seat is on a horizontal line through the axle of the wheel e Greek letter (theta) o en stands for the measure of an angle through which an object rotates A wheel rotates through 360deg each revolution so is not restricted If you plot in degrees on the horizontal axis and the height above the ground y in meters on the vertical axis the graph looks like Figure 5-1b Notice that the graph has repeating y-values corresponding to each revolution of the Ferris wheel

90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20

Figure 5-1a Figure 5-1b

Find the function that corresponds to the graph of a sinusoid and graph it on your grapher

Introduction to Periodic FunctionsAs you ride a Ferris wheel your distance from the ground depends on the number

5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective

1 e graph in Figure 5-1c is the sine function (pronounced ldquosignrdquo) Its abbreviation is sin and it is written sin( ) or sin Plot f 1 (x) sin(x) on your grapher (using x instead of ) Use the window shown and make sure your grapher is in degree mode Does your graph agree with the gure

20

2720deg540deg360deg180deg90deg

y

x

Figure 5-1c

2 e graphs in Figures 5-1b and 5-1c are called sinusoids (pronounced like ldquosinusrdquo a skull cavity) What two transformations must you perform on the parent sine function in Figure 5-1c to get the sinusoid in Figure 5-1b

3 Enter into your grapher an appropriate equation for the sinusoid in Figure 5-1b as f 2 (x) Verify that your equation gives the correct graph

4 Explain how an angle can have a measure greater than 180deg Explain the real-world signi cance of the negative values of and x in Figures 5-1b and 5-1c

246 Chapter 5 Periodic Functions and Right Triangle Problems

In Chapters 1ndash4 you studied various types of functions and how these functions can be mathematical models of the real world In this chapter you will study functions for which the y-values repeat at regular intervals You will study these periodic functions in four ways

cos u

__ r displacement of adjacent leg

_______________________ length of hypotenuse

( is the Greek letter theta)

y cos

0deg 1

30deg 08660

60deg 05

90deg 0

is is the graph of a cosine function Here y depends on the angle which can take on negative values and values greater than 180deg 1

1y

180deg 180deg 360deg 540deg 720deg

The trigonometric functions cosine sine tangent cotangent secant and cosecant are initially defined as ratios of sides of a right triangle The definitions are extended to positive and negative angles measuring rotation by forming a reference right triangle whose legs are positive or negative displacements and whose hypotenuse is the radius of the circle formed as the angle increases The resulting functions are periodic as the angle increases beyond 360deg

ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY

In Chapters 1ndash4 you studied various types of functions and how

Mathematical Overview

v

(u v)

rv

uu


S e c t i o n 5 -1 PL AN N I N G

Class Time1__2day

Homework AssignmentProblems1ndash4

Teaching ResourcesExploration5-1aTransformedPeriodic

Functions

Technology Resources

ActivityASineWaveTracer

TE ACH I N G

Important Terms and ConceptsPeriodicfunctionsTh eta(u)SinefunctionDegreemode(calculatorsetting)SinusoidsParentsinefunction

Section Notes

Th issectionintroducesthesinefunctionandtheconceptofangleasameasureofrotationItlaysthefoundationforusingtrigonometricfunctionsasmathematicalmodelsYoucanassignSection5-1ashomeworkonthenightoftheChapter4testorasagroupactivitytobecompletedinclassNoclassroomdiscussionisneededbeforestudentsbegintheactivity

InthistextsinusoidalfunctionsarefirstintroducedusingdegreesbecausestudentsarefamiliarwithmeasuringanglesindegreesRadianmeasureisintroducedinSection6-4whenstudentsaremorecomfortablewithtrigonometricfunctions




90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes

InExploration 5-1astudentsaregivenapre-imagegraphandatransformedgraphofaperiodicfunctionTh eyareaskedtofindtheequationofthetransformationAllowabout20minutestocompletethisexploration

Technology Notes

ActivityASineWaveTracerintheInstructorrsquos Resource BookallowsstudentstoconstructasinewavebytracingapointastwopointsareanimatedTh etracerisconstructedbasedonradiananglemeasurebutitcanbeadaptedasabasicintroductiontothesinefunctionanditsgraphTh isactivityisforintermediateSketchpadusersandwilltake30ndash40minutes

PRO B LE M N OTES

Problem 1 introducesthesinefunction1 Th egraphshouldmatchFigure5-1c

Problem 2requiresstudentstorecalltheworktheydidwithtransformationsinChapter12 Verticaldilationby9verticaltranslationby113 f2(x)51119sin(x)

Problem 4asksstudentstothinkaboutangleswithnegativemeasuresandwithmeasuresgreaterthan1804 AnswerswillvaryTh eanglemeasureshowmuchsomethinghasrotatedItcanrotatemorethan360bycontinuingtorotateafterithasrotatedafullcircleItcanalsorotateintheotherdirection

249Section 5-2 Measurement of Rotation

e same position can have several corresponding angle measures For instance the 493deg angle terminates in the same position as the 133deg angle a er one full revolution (360deg) more e 227deg angle terminates there as well by rotating clockwise instead of counterclockwise Figure 5-2c shows these three coterminal angles

v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg

227deg 133deg 360deg( 1) Figure 5-2c

Letters such as may be used for the measure of an angle or for the angle itself Other Greek letters are o en used as well (alpha) (beta) (gamma) (phi) (pronounced ldquofyerdquo or ldquofeerdquo) and (omega)

You might recognize some of the Greek letters on this subway sign in Athens Greece

DEFINITION Coterminal AnglesTwo angles in standard position are coterminal if and only if their degree measures di er by a multiple of 360deg at is and are coterminal if and only if

360degn

where n stands for an integer

Note Coterminal angles have terminal sides that coincide hence the name

To draw an angle in standard position you can nd the measure of the positive acute angle between the horizontal axis and the terminal side is angle is called the reference angle


Measurement of RotationIn the Ferris wheel problem of Section 5-1 you saw that you can use an angle to measure an amount of rotation In this section you will extend the concept of an angle to angles whose measures are greater than 180deg and to angles whose measures are negative You will learn why functions such as your height above the ground are periodic functions of the angle through which the Ferris wheel turns

Given an angle of any measure draw a picture of that angle

An angle as a measure of rotation can be as large as you like For instance a gure skater might spin through an angle of thousands of degrees To put this idea into mathematical terms consider a ray with a xed starting point Let the ray rotate through a certain number of degrees and come to rest in a terminal (or nal) position as in Figure 5-2a

So that the terminal position is uniquely determined by the angle measure a standard position is de ned e initial position of the rotating ray is along the positive horizontal axis in a coordinate system with its starting point at the origin Counterclockwise rotation to the terminal position is measured in positive degrees and clockwise rotation is measured in negative degrees

DEFINITION Standard Position of an AngleAn angle is in standard position in a Cartesian coordinate system if

counterclockwise from the horizontal axis if the angle measure is positive and clockwise from the horizontal axis if the angle measure is negative

Figure 5-2b shows a rotating ray in several positions in a uv-coordinate system (v for vertical) with a point (u v) on the ray at a xed distance from the origin e angle in standard position measures the location of the ray ( e customary variables x and y will be used later for other purposes)

v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b

Measurement of RotationIn the Ferris wheel problem of Section 5-1 you saw that you can use an angle

5 -2

Given an angle of any measure draw a picture of that angleObjective

v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


0and360instandardpositionandthencalculatethemeasureofthereferenceangle

Example2onpage251showshowtofindthereferenceangleforananglegreaterthan360byfirstfindingacoterminalanglewithmeasurebetween0and360Helpstudentsrealizethattheycanfindthenumberofwholerevolutionsbydividingtheanglemeasureby360Th entofindthecoterminalangletheycaneithermultiplythedecimalpartofthequotientby360

orsubtract360timesthenumberofwholerevolutionsfromtheoriginalanglemeasure

MakesurestudentsunderstandthatthereferenceangleisalwaysapositiveacuteanglethatisldquonestledagainstrdquothehorizontalaxisBecauseareferenceanglemustbeacuteanglesof090180and270andanglescoterminalwiththesefouranglesdonothavereferenceangles


Class Time1day

Homework AssignmentRAQ1ndashQ10Problems159192125

272930

Teaching ResourcesExploration5-2ReferenceAngles

Technology Resources CASActivity5-2aMeasurement

Conversions

TE ACH I N G

Important Terms and ConceptsInitialpositionTerminalpositionStandardpositionCounterclockwiseClockwiseCoterminalangles(5u1360n)ReferenceangleDegreesminutesseconds

Section Notes

IfyoudidnotassignSection5-1ashomeworkaftertheChapter4testyoucancoverSections5-1and5-2onthesamedaySection5-1isagoodgroupactivity

Th emainpointtogetacrossinthissectionisthatanglescanhavemeasuresoutsidetherangeof0to180Toillustrateyoumightpresentthefamiliarexampleofthenumberofdegreesrotatedbythehandsofaclock

ItisworthspendingafewminutesmakingsurestudentsarecomfortablepronouncingandwritingtheGreeklettersoftenassociatedwithangles

Example1onpage251illustrateshowtosketchangleswithmeasuresbetween



v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


Emphasizethatareferenceangleisnever measuredbetweentheverticalaxisandtheterminalsideoftheangleAlsostressthatreferenceanglesalwaysgoinacounterclockwisedirectionTh ussomereferenceanglesgofromthehorizontalaxistotheterminalsidewhereasothersgofromtheterminalsidetothehorizontalaxis

Eventhebeststudentssometimesdrawandcalculatereferenceanglesincorrectly

ReferencetrianglesareintroducedinSection5-3

Th issectionpresentsanexcellentopportunitytoreviewhowtosetthecalculationmodesonstudentsrsquographersForexamplebyfactorydefaulttheTI-NspirersquosangleoperationsaresettoradianmodeTochangethisgotoSystem SettingstabdowntotheAngle downmenupressdownonthecursorwheelandselectDegreePressENTER

twoorthreetimestoselectthenewmodeandapplyittoyourcurrentdocument

YoucanalwaysforceaTI-NspiregraphertocomputeinanyanglemoderegardlessofthemodeitissetinAlthoughstudentsarenotreadytocomputetrigonometricfunctionvaluesinthissectionthefigureshowsthatevenwiththeTI-Nspiresetindegreemodesin(__3radians)iscorrectlycomputed(notethesmallrfollowingtheanglemeasureinthefigure)PressCTRL+CATALOG tofindtheradian(r)anddegree(deg)symbolstooverridethesystemsettingsStudentsfindthisunit-overridefeatureespeciallyhelpful

Diff erentiating Instructionbull PassoutthelistofChapter5

vocabularyavailableatwwwkeypresscomkeyonlineforstudentstolookupandtranslateintheirbilingualdictionaries

bull Havestudentsexplaintheconceptofreference anglesintheirjournalsincludingvisualexamples

bull Providevisualdefinitionsforwordssuchasacute obtuse opposite andadjacentthatmayappearinbilingualdictionarieswithouttheirmathematicalmeaningsELLstudentsshouldwritethesedefinitionsintheirjournalsandintheirbilingualdictionaries

bull SpecificallyintroducetheconceptofdegreesminutesandsecondsSomestudentshaveonlyseendegreesdividedintodecimalparts


Example 1 shows how to nd reference angles for angles terminating in each of the four quadrants

Sketch angles of 71deg 133deg 254deg and 317deg in standard position and calculate the measure of each reference angle

To calculate the measure of the reference angle sketch an angle in the appropriate quadrant then look at the geometry to gure out what to do

Figure 5-2d shows the four angles along with their reference angles For an angle between 0deg and 90deg (in Quadrant I) the angle and the reference angle are the same For angles in other quadrants you have to calculate the positive acute angle between the u-axis and the terminal side of the angle

v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg

ref 180deg 133deg 47degv

u 254deg

ref 74deg

ref 254deg 180deg 74deg

317deg ref 43deg

v

u


Figure 5-2d

Note that if the angle is not between 0deg and 360deg you can rst nd a coterminal angle that is between these values It then becomes an ldquooldrdquo problem like Example 1

Sketch an angle of 4897deg in standard position and calculate the measure of the reference angle

4897 ____ 360 136027 Divide 4897 by 360 to nd the number of whole revolutions

is number tells you that the terminal side makes 13 whole revolutions plus another 06027 revolution To nd out which quadrant the angle falls in multiply the decimal part of the number of revolutions by 360 to nd the number of degrees e answer is c a coterminal angle to between 0deg and 360deg

c (06027)(360) 217deg Compute without rounding


EXAMPLE 1


SOLUTION


EXAMPLE 2

4897____360 SOLUTION


DEFINITION Reference Angle e reference angle of an angle in standard position is the positive acute angle between the horizontal axis and the terminal side

Note Reference angles are always measured counterclockwise Angles whose terminal sides fall on one of the axes do not have reference angles

In this exploration you will apply this de nition to nd the measures of several reference angles

1 e gure shows an angle 152deg in standard position e reference angle ref is measured counterclockwise between the terminal side of and the nearest side of the horizontal axis Show that you know what reference angle means by drawing ref and calculating its measure

v

u152deg

2 e gure shows 250deg Sketch the reference angle and calculate its measure

v

u250deg

3 You should have drawn arrowheads on the arcs for the reference angles in Problems 1 and 2 If you havenrsquot draw them now Explain why the arc for 152deg goes from the terminal side to the u-axis but the arc for 250deg goes from the u-axis to the terminal side

4 Amos Take thinks the reference angle for 250deg should go to the v-axis because the terminal side is closer to it than the u-axis Tell Amos why his conclusion does not agree with the de nition of reference angle in Problem 1

5 Sketch an angle of 310deg in standard position Sketch its reference angle and nd the measure of the reference angle

6 Sketch an angle whose measure is between 0deg and 90deg What is the reference angle of this angle

7 e gure shows an angle of 150deg Sketch the reference angle and nd its measure

v

u

150deg2

8 e gure in Problem 7 shows a point 2 units from the origin and on the terminal side of the angle Draw a segment from this point perpendicular to the u-axis thus forming a right triangle whose hypotenuse is 2 units long Use what you recall from geometry to nd the lengths of the two legs of the triangle

9 What did you learn as a result of doing this exploration that you did not know before

1 e gure shows an angle 152deg in 4 Amos Take thinks the reference angle for 250deg

E X P L O R AT I O N 5 -2 R e f e r e n c e A n g l e s


6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u

Differentiating Instruction (continued)bull Studentsmighthavebeenintroduced

totrigonometricfunctionsfromaperspectiveotherthanthetriangletrigonometryoftenintroducedintheUnitedStatesStudentswillbenefitfromtheuseofconstantvisualreferences

Exploration Notes

Exploration 5-2 isashortandimportantexerciseonreferenceanglesForProblem8studentsmustrecalltherelationshipsamongthesidesofa30-60-90triangleRefertotheSectionNotesforcommentsonreferenceanglesAllowabout15minutes1 uref51802152528

28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u

3 Becausetheanglemustbecounterclockwisesothatitsmeasurewillbepositive4 Becauseitmustgotothenearestsideofthehorizontalaxis

5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes

CASActivity5-2aMeasurementConversionsintroducesstudentstotheuseofunitconversionsonaCASandillustratesthelinearrelationshipsbetweenmanyrelatedunitsofmeasureAllow20ndash25minutes

CAS Suggestions

StudentsmayfindithelpfultodotheunitconversionsinthissectiononaCASwheretheycanbehandledlikeanyothervariablesAfulllistingofunitabbreviationsisavailableintheCatalogbutfewstudentsneedtorefertothismenubecauseoftheirfamiliaritywithunitsfromtheirsciencecoursesToindicatethatunitsaretobeappliedinsertanunderscorecharacterbeforetheunitabbreviationTh eunderscoreandconversioncharacterscanbefoundbypressing CTRL+CATALOG

Th efigureshowssomeconversionsbetweenfamiliarlengthunitsincludingadditionoflengthvaluesofdifferentunits(Th edefaultoutputunitsystemcanbechangedbyvisitingtheSystem

Settings menu)NoticethattheTI-Nspiresyntaxforsecondsis_s(not_sec)FinallytheTI-Nspireautomaticallyconvertscomplexunitsintotheircompositeform(eg1kgm_____s isequivalentto1newton)

8 Duplicatingthetriangleaboveitselfmakesanangleof60ateachvertexsothelargetriangleisequiangularandthereforeequilateralSoallsidesareoflength2andtheleft(vertical)legoftheoriginaltriangleishalfof2or1(21becauseitisbelowthehorizontalaxis)Sotheother(horizontal)legis

______2221252

__3(2

__3becauseit

istotheleftoftheverticalaxis)

30deg1

12

2

3

v

u

9 Answersmayvary

Section 5-2 Measurement of Rotation


13 986deg 14 573deg 15 1541deg 16 2732deg 17 5481deg 18 7321deg 19 2746deg 20 3614deg

For Problems 21ndash26 the angles are measured in degrees minutes and seconds ere are 60 minutes (60 ) in a degree and 60 seconds (60 ) in a minute To nd 180deg 137deg24 you calculate 179deg60 137deg24 Sketch each angle in standard position mark the reference angle and nd its measure

21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44

For Problems 27 and 28 sketch a reasonable graph of the function showing how the dependent variable is related to the independent variable 27 A student jumps up and down on a trampoline

Her distance from the ground depends on time

28 e pendulum in a grandfather clock swings back and forth e distance from the end of the pendulum to the le side of the clock depends on time

For Problems 29 and 30 write an equation for the image function g (solid) in terms of the pre-image function f (dashed) 29

30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2

Sketch the 217deg angle in Quadrant III as in Figure 5-2e

ref 37deg

c 217deg

v

u 4897deg

How many revolutions

Where will it end up

v

u

Figure 5-2e

From the gure you should be able to see that


As you draw the reference angle remember that it is always between the terminal side and the horizontal axis (never the vertical axis) e reference angle sometimes goes from the axis to the terminal side and sometimes from the terminal side to the axis To gure out which way it goes recall that the reference angle is positive us it always goes in the counterclockwise direction

Reading Analysis

From what you have read in this section what do you consider to be the main idea How can an angle have a measure greater than 180deg or a negative measure If the terminal side of an angle is drawn in standard position in a uv-coordinate system why can there be more than one value for the measure of the angle but only one value for the measure of the reference angle

Quick Review Q1 A function that repeats its values at regular

intervals is called a function

In Problems Q2ndashQ5 describe the transformation Q2 g(x) 5f (x) Q3 g(x) f (3x) Q4 g(x) 4 f (x) Q5 g(x) f (x 2) Q6 If f (x) 2x 6 then f 1 (x)

Q7 How many degrees are there in two revolutions

Q8 Sketch the graph of y 2 x Q9 40 is 20 of what number

Q10 x 20 ___ x 5

A x 15 B x 4 C x 25 D x 100 E None of these

For Problems 1ndash20 sketch the angle in standard position mark the reference angle and nd its measure 1 130deg 2 198deg 3 259deg 4 147deg 5 342deg 6 21deg 7 54deg 8 283deg 9 160deg 10 220deg 11 295deg 12 86deg

5min


PRO B LE M N OTES

Q1 Periodic Q2 y-dilationby5Q3 x-dilationby1__3Q4 y-translationby14Q5 x-translationby12Q6 y51__2x23 Q7 720Q8

Q9 200 Q10 A

Problems 1ndash20 requirestudentstosketchanglesinstandardpositionandfindthemeasuresofthereferenceangles1 uref550 2 uref518

v

uref

v

uref

3 uref579 4 uref533v

u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253

Problems 21ndash26involvearithmeticwithanglemeasuresgivenindegreesminutesandsecondsStudentsmayneedtoreviewhowtoconvertamongtheseunitsSomegraphershaveafeaturethatconvertsanglemeasuresfromdegreesminutesandsecondstodegreeswithdecimalsHoweverstudentsshouldbeabletodotheseconversionswithoutthisfeature21 uref53423 22 uref58829

v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref

Problems 27 and 28 askstudentstosketchreasonablegraphsforsituationsthatexhibitperiodicbehavior

27

Problems 29 and 30reviewthetransformationsstudiedinChapter1andpreparestudentsforthenextsection29 g(x)541f(x21)30 g(x)53f   x __2

Seepage999foranswerstoProblems17ndash20and28

Distance

Time13 uref5814 14 uref5573v

uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref


255Section 5-3 Sine and Cosine Functions

Periodicity is common e phases of the moon are one example of a periodic phenomenon

DEFINITION Periodic Function e function f is a periodic function of x if and only if there is a number p 0 for which f (x p) f (x) for all values of x in the domain

If p is the smallest such number then p is called the period of the function

De nition of Sine and Cosine for Any AngleTo understand why sine and cosine are periodic functions consider a ray rotating about the origin in a uv-coordinate system forming a variable angle in standard position e le side of Figure 5-3c shows the ray terminating in Quadrant I A point on the ray r units from the origin traces a circle of radius r as the ray rotates e coordinates (u v) of the point vary but r stays constant

v(u v)

u

v

u

Radius r Terminalside of

Draw aperpendicular

Ray rotates

Referencetriangle

Hypotenuse (radius) r (u v)

Vertical leg v (opposite )

Horizontal leg u (adjacent to )

v

u

Figure 5-3c

Drawing a perpendicular from point (u v) to the u-axis forms a right triangle with as one of the acute angles is triangle is called the reference triangle As shown on the right in Figure 5-3c the coordinate u is the length of the leg adjacent to angle and the coordinate v is the length of the leg opposite angle e radius r is the length of the hypotenuse

e right triangle de nitions of the sine and cosine functions are

sin opposite leg

__________ hypotenuse cos adjacent leg

__________ hypotenuse

ese functions are called trigonometric functions from the Greek roots for ldquotrianglerdquo (trigon-) and ldquomeasurementrdquo (-metry)


Sine and Cosine FunctionsFrom previous mathematics courses you may recall working with sine cosine and tangent for angles in a right triangle Your grapher has these functions in it If you plot the graphs of y sin and y cos you get the periodic functions shown in Figure 5-3a Each graph is called a sinusoid as you learned in Section 5-1 To get these graphs you may enter the equations in the form y sin x and y cos x and use degree mode

y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a

In this section you will see how the reference angles of Section 5-2 let you extend the right triangle de nitions of sine and cosine to include angles of any measure You will also see how these de nitions lead to sinusoids

Extend the de nitions of sine and cosine to any angle

A periodic function is a function whose values repeat at regular intervals e graphs in Figure 5-3a are examples e part of the graph from any point to the point where the graph starts repeating itself is called a cycle For a periodic function the period is the di erence between the horizontal coordinates corresponding to one cycle as shown in Figure 5-3b e sine and cosine functions complete a cycle every 360deg as you can see in Figure 5-3a So the period of these functions is 360deg A horizontal translation of one period makes the pre-image and image graphs identical

One period

Equal y-valuesOne cycle

Figure 5-3b

Sine and Cosine FunctionsFrom previous mathematics courses you may recall working with sine

5 -3

Extend the de nitions of sine and cosine to any angleObjective



Class Time1day

Homework AssignmentRAQ1ndashQ10Problems1ndash23odd

Teaching ResourcesExploration5-3aDefinitionsofSineand

CosineExploration5-3buv-Graphsand

uy-GraphsofSinusoidsExploration5-3cParentSinusoidsBlacklineMasters Example3Problems15ndash20 TrigonometricRatiosTable


SineWaveTracer

Exploration5-3buv-Graphsanduy-GraphsofSinusoids

TE ACH I N G

Important Terms and ConceptsPeriodicfunctionCyclePeriodTrigonometricfunctionsDisplacementRighttriangledefinitionsofsineand

cosineSinefunctionofanysizeangleCosinefunctionofanysizeangleSinusoidParentsinefunctionCriticalpoints






v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes

InpreviouscoursesstudentslearnedhowtofindthesineandcosineoftheacuteanglesinarighttriangleInthissectionthedefinitionsofsineandcosineareextendedtoanglesofanymeasuremakingitpossibletodiscussy5sinu andy5cosuasperiodicfunctionscalledsinusoidsItisimportanttotaketimetomakesurestudentsunderstandthatthedefinitionsarebeingextendedtoanglesofanymeasureOtherwisetheywillbeforeverweddedtothetriangledefinitionsandmayhavetroubleworkingwithnon-acuteangles

Th egoalsofthissectionaretohavestudentsunderstandandlearnthedefinitionsofthesineandcosinefunctionsandtounderstandandlearntheparentgraphsofthesineandcosinefunctions

ForthissectionandthroughouttheremainderofthischaptergraphersshouldbeindegreemodeSomegraphersuseradianmodeasthedefaultsettingShowstudentshowtochangetheirgrapherstodegreemodeandremindthemtocheckthemodeeachtimetheyturnonthegrapherorclearthememoryAlsonotethatalthoughu isusedtorepresentanglesmeasuredindegreesstudentswillneedtographequationsusingxastheindependentvariable

Tointroducethetopicofperiodicfunctionshavestudentsgraphthefunctiony5sinxontheirgraphersTh enhavethemtracethefunctionandobservethatthey-valuesrepeatevery360Usethegraphtoexplaintheconceptsofcycleandperiod

Section 5-3 Sine and Cosine Functions


Periodicity of Sine and CosineFigure 5-3g shows why the sine function is periodic On the le is shown as an angle in standard position in a uv-coordinate system As increases from 0deg v is positive and increasing until it equals r when reaches 90deg en v decreases becoming negative as passes 180deg until it equals r when reaches 270deg From 270deg to 360deg v is negative but increasing until it equals 0 Beyond 360deg the pattern repeats Recall that sin v _ r us sin starts at 0 increases to 1 decreases to

1 and then increases to 0 as increases from 0deg through 360deg e right side of Figure 5-3g shows as the horizontal coordinate in a y-coordinate system with y sin is is the rst sinusoid of Figure 5-3a

y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating

90deg 0deg 90deg 180deg 270deg 360deg

Figure 5-3g

You can see how the rotating ray in a uv-coordinate system generates a sinusoid by viewing the Sine Wave Tracer exploration at wwwkeymathcomprecalc You can also create your own animation using geometry so ware such as e Geometerrsquos Sketchpadreg

Draw angle equal to 147deg in standard position in a uv-coordinate system Draw the reference triangle and show the measure of ref the reference angle Find cos 147deg and cos ref and explain the relationship between the two cosine values

Draw the 147deg angle and its reference angle as in Figure 5-3h Pick a point on the terminal side of the angle and draw a perpendicular to the horizontal axis forming the reference triangle

ref 180deg 147deg 33deg Because ref and 147deg must sum to 180deg

cos 147deg 08386

cos 33deg 08386 By calculator

Both cosine values have the same magnitude Cos 147deg is negative because the horizontal coordinate of a point in Quadrant II is negative e radius r is always considered to be positive because it is the radius of a circle traced as the ray rotates

Note When you write the cosine of an angle in degrees such as cos 147deg you must write the degree symbol Writing cos 147 without the degree symbol has a di erent meaning as you will see when you learn about angles in radians in the next chapter

v

u

v is negative decreasing

v is positive increasing

v is positive decreasing

v is negative increasing

Draw angle the reference triangle and show the measure of

EXAMPLE 1

Draw the 147deg angle and its reference angle as in Figure 5-3h Pick a point on the terminal side of the angle and draw a perpendicular to the horizontal axis

SOLUTION

v

u33deg147deg

Figure 5-3h


As increases beyond 90deg the values of u v or both become negative As shown in Figure 5-3d the reference triangle appears in dierent quadrants with the reference angle at the origin Because u and v can be negative consider them to be displacements of the point (u v) from the respective axes e value of r stays positive because it is the radius of the circle e denitions of sine and cosine for any angle use these displacements u and v

u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)

in Quadrant IVFigure 5-3d

DEFINITION Sine and Cosine Functions for Any AngleLet (u v) be a point r units from the origin on a rotating ray Let be the angle to the ray in standard position en

sin v __ r

vertical displacement __________________ radius cos

u __ r

horizontal displacement ____________________ radius

You can remember these denitions by thinking ldquov as in verticalrdquo and ldquou comes before v in the alphabet like x comes before yrdquo With respect to the reference triangle v is the displacement opposite the reference angle and u is the displacement adjacent to it e radius is always the hypotenuse of the reference triangle

Figure 5-3e shows the signs of the displacements u and v for angle in the four quadrants

Note that the values of sin and cos depend only on the measure of angle not on the location of the point on the terminal side

As shown in Figure 5-3f reference triangles for the same angle are similar us

sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u

Radius is always positive

Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f


Section Notes (continued)

WhensineandcosineareconsideredasfunctionsrighttriangledefinitionsofsineandcosineareextendedReviewtherighttriangledefinitionsofsineandcosinewithstudentsTh endiscussthematerialinthetextthatexplainshowthedefinitionscanbeextendedtoanglesofanymeasureNotethatthereferenceangledeterminesareferencetriangletowhichtherighttriangledefinitionsapplyMakesurestudentsunderstandthat(u v)isanypointontheterminalsideoftheangleandristhedistancefrom(u v)totheoriginEmphasizethatrisalwayspositivebecauseitisadistancebutuandvcanbeeitherpositiveornegativeTh ereforethesineorcosineofananglecanbepositiveornegative

PointouttostudentsthatcoterminalangleshavethesamesinevaluesandthesamecosinevaluesbecausetheyhavethesameterminalpositionsTh isexplainswhythesineandcosinefunctionsareperiodicwithcyclesthatrepeatevery360

Example1onpage257illustratestherelationshipbetweenthecosineofanangleandthecosineofthecorre-spondingreferenceangleBesuretodiscussthenoteafterExample1whichemphasizestheimportanceofincludingthedegreesignwhendenotingthesineorcosineofanangleindegreesIfthedegreesignisnotincludedtheangleisassumedtobeinradiansSoforexampletodenotethecosineofananglewithmeasure147studentsshouldwritecos147notcos147

Example2onpage258demonstrateshowtoapplythedefinitionsofsineandcosinetofindthesineandcosineofananglebasedonapointontheterminalsideoftheangle

Example3onpage258showshowtographatransformationofthesinefunctionbyplottingcriticalpointsAblacklinemasterofFigure5-3jisavailableintheInstructorrsquos Resource Book

ConsidergivingstudentsacopyoftheTrigonometricRatiosTablefromtheBlacklineMasterssectionintheInstructorrsquos Resource BookHavestudentskeepthetableinafolderforreferenceTh eycancontinuetofillinvaluesastheylearnaboutmore

trigonometricratiosinSection5-4andwhentheylearnaboutradianmeasureinSection6-4

StudentswhouseTI-NspiregrapherscandynamicallyrecreateExample3inaGraphsampGeometrywindowChangethewindowsettingsto[2800800]forxwithscale30and[266]forywithscale1Turnthegridongraphf(x)5sinxandmovethecursortothefirstrelativemaximum




y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257

Differentiating Instructionbull Becausetheymayhaveapproached

trigonometricfunctionsotherthanthroughrighttrianglessomestudentswillneedahandoutdefiningthetrigonometricfunctionsintermsofarighttriangleandrelatingthesedefinitionstoanillustration

bull Makesurestudentsconnectsine andcosinewiththeirabbreviationssin andcosInmanylanguagesthesefunctionsarepronouncedldquoseenusrdquoandldquocoseenusrdquoThismaymakeitdifficultforELLstudentstoconnecttheEnglishpronunciationwiththeirabbreviations

bull Havestudentswriteintheirownwordsdefinitionsfor sine cosine andperiod intheirjournals

bull HaveELLstudentdotheReadingAnalysisinpairs

Exploration Notes

IfyouhavetimeforonlyoneexplorationitisrecommendedthatyouuseExploration 5-3a

Exploration 5-3aconnectstherighttriangledefinitionsofsine andcosine totheextendeddefinitionsandcanbeusedinplaceofExample1ItrequiresstudentstousearulerandaprotractortomakeaccuratemeasurementsAllow15ndash20minutesforstudentstocompletetheexploration

Exploration 5-3bdemonstratestherelationshipbetweenaunitcircleandthegraphsofy5sinxandy5cosxTheexplorationtakesabout 20minutesYoucanusethisexplorationwithSection6-4insteadofwithSection5-3ifyouprefer

Exploration 5-3cexploresthegraphandtransformationsoftheparentfunctiony5sinxIfyouworkthroughtheproblemswiththeclassasawholethisexplorationcanreplaceExample3

pointtotherightofthey-axisWhenthecursorchangestoanXwitharrowsclickandholdtodragthepointThischangesboththesinusoidanditsequation(bothonthescreenandinthecommandline)asstudentsdragthepointThiscanbeadramaticwaytoemphasizetransformationsoftrigonometricfunctions



y

1360deg

Mark high low and middle points

Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis

From what you have read in this section what do you consider to be the main idea Give a real-world example of what it means for a function to be periodic How are the de nitions of sine and cosine extended from right triangles with angle measures limited to between 0deg and 90deg to angles of any measure positive or negative What is the di erence between the way appears in a uv-coordinate system and the way it appears in a y-coordinate system

Quick Review Q1 Write the general equation for an exponential

function

Q2 e equation y 3 x 12 represents a particular function

Q3 Find the reference angle for a 241deg angle

Q4 Name these Greek letters

Q5 What transformation of the pre-image function y x 5 is the image function y (x 3) 5

Q6 Find x if 5 log 2 log x

Q7 Sketch a reasonable graph showing the height of your foot above the pavement as a function of the distance your bicycle has traveled

Q8 3 7 0 (37 with a zero exponent not 37 degrees)

Q9 What is the value of 5 ( ve factorial)

Q10 What percent of 300 is 60

For Problems 1ndash6 sketch the angle in standard position in a uv-coordinate system Draw the reference triangle showing the measure of the reference angle Find the sine or cosine of the angle and its reference angle Explain the relationship between them 1 sin 250deg 2 sin 320deg 3 cos 140deg 4 cos 200deg 5 cos 300deg 6 sin 120deg

5min

Reading Analysis Q7 Sketch a reasonable graph showing the height of

Problem Set 5-3


e terminal side of angle contains the point (u v) (8 5) Sketch the angle in standard position Use the de nitions of sine and cosine to nd sin and cos

As shown in Figure 5-3i draw the point (8 5) and draw angle with its terminal side passing through the point Draw a perpendicular from the point (8 5) to the horizontal axis forming the reference triangle Label the displacements u 8 and v 5 Calculate the radius r using the Pythagorean theorem and show it on your sketch

r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____

___ 89 05299 Sine is opposite displacement

______________ hypotenuse

cos 8 _____

___ 89 08479 Cosine is adjacent displacement


Figure 5-3j shows the parent sine function y sin You can plot sinusoids with other proportions and locations by transforming this parent graph Example 3 shows you how to do this

y

1

360deg

Figure 5-3j

Let y 4 sin What transformation of the parent sine function is this On a copy of Figure 5-3j sketch the graph of this image sinusoid Check your sketch by plotting the parent sinusoid and the transformed sinusoid on the same screen

e transformation is a vertical dilation by a factor of 4

Find places where the pre-image function has high points low points or zeros Multiply the y-values by 4 and plot the resulting points (Figure 5-3k on the next page le side) Sketch a smooth curve through the critical points (Figure 5-3k right side) Your grapher will con rm that your sketch is correct

e terminal side of angle in standard position Use the de nitions of sine and cosine to nd sin

EXAMPLE 2

As shown in Figure 5-3iside passing through the point Draw a perpendicular from the point (8

SOLUTION

v

u8

5__89

Figure 5-3i

Let y copy of Figure 5-3j sketch the graph of this image sinusoid Check your sketch by

EXAMPLE 3

e transformation is a vertical dilation by a factor of 4SOLUTION


Technology Notes

Th ediscussionbelowFigure5-3gsuggestsusingtheSineWaveTracerexplorationatwwwkeymathcomprecalc Th eexplorationallowsstudentstoinvestigatepropertiesofthesinusoidalwaveusinganinteractivetracerItisalsousedinSection5-4Problem45

Exploration 5-3buv-Graphsanduy-GraphsofSinusoidsintheInstructorrsquos Resource Book canbedonewiththehelpofSketchpad

CAS Suggestions

Th edefinitionofperiodicfunctionscanbeusedonaCAStoverifyamultipleoftheperiodoftrigonometricfunctionsTh efirsttwolinesofthefigureshowthisforsineDonotconfusethiswithverificationoftheperioditselfACAStestswhethertheaddedvalueofpisamultipleoftheperiodLine3ofthefigureshowsthat180degisnotamultipleoftheperiodofsinebutline4showsthat720degisTh isisausefultoolforverifyingthatagivenperiodmightbecorrect

IfyourstudentsarefamiliarwithvectorsfrompreviousmathstudiesorasciencecourseanalternativewaytodeterminethesineandcosinevaluesinExample2istodefinecosineandsineasthecorrespondingx-andy-coordinatesofaunitvectorinthesamedirectionasagivenvectorTh eCAScommandforthisisunitV(Notethatsquarebracketsareusedtoindicatevectors)

Q1 y5ab x a 0 b 0Q2 PowerQ3 61Q4 AlphabetagammaphiQ5 x-translationby13Q6 x532


y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES

Problems 1ndash6exploretherelationshipbetweenthesineandcosineofanangleandthesineandcosineofthecorrespondingreferenceangleProblem 23followsdirectlyfromtheworkontheseproblemsIfyougavestudentstheTrigonometricRatiosTablefromtheInstructorrsquos Resource Bookyoumightaskthemtorefertothetablewhilecompletingtheseproblems2 uref540

v

uref

sin3205206427sin40506427sin32052sin403 uref540

v

uref

cos1405207660cos40507660cos14052cos404 uref520

v

u

ref

cos2005209396cos20509396cos20052cos20

Seepage999foranswerstoProblems5and6

Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance

261Section 5-4 Values of the Six Trigonometric Functions

Values of the Six Trigonometric FunctionsIn Section 5-3 you recalled the de nitions of the sine and cosine of an angle and saw how to extend these de nitions to include angles beyond the range of 0deg to 90deg With the extended de nitions y sin and y cos are periodic functions whose graphs are called sinusoids In this section you will de ne four other trigonometric functions You will learn how to evaluate all six trigonometric functions approximately by calculator and exactly in special cases using the de nitions In the next section you will use what you have learned to calculate unknown side lengths and angle measures in right triangles

Be able to nd values of the six trigonometric functions approximately by calculator for any angle and exactly for certain special angles

Sine and cosine have been de ned for any angle as ratios of the coordinates (u v) of a point on the terminal side of the angle and equivalently as ratios of the displacements in the reference triangle

sin v __ r

vertical displacement __________________ radius

opposite __________ hypotenuse

cos u

__ r horizontal displacement

____________________ radius adjacent


In this exploration you will explore the values of sine and cosine for various angles

Values of the Six Trigonometric Functions

5 - 4


Objective

E X P L O R AT I O N 5 - 4 V a l u e s o f t h e S i n e a n d C o s i n e Fu n c t i o n s 1 is gure shows an angle in standard

position in a uv-coordinate system Write the sine and cosine of in terms of the coordinates (u v) and the distance r from the origin to the point

u

r

v(u v)

2 is gure shows an angle of 123deg in standard position in a uv-coordinate system By calculator nd sin 123deg and cos 123deg Write the answers in ellipsis format Explain why sin 123deg is positive but cos 123deg is negative

u

r

v

(u v)

123deg

continued


For Problems 7ndash14 use the de nitions of sine and cosine to write sin and cos for angles whose terminal side contains the given point 7 (7 11) 8 (4 1) 9 ( 2 5) 10 ( 6 9) 11 (4 8) 12 (8 3) 13 ( 24 7) (What do you notice about r) 14 ( 3 4) (What do you notice about r)

Figure 5-3l shows the parent function graphs y sin and y cos For Problems 15ndash20 give the transformation of the parent function represented by the equation Sketch the transformed graph on a copy of Figure 5-3l Con rm your sketch by plotting both graphs on the same screen on your grapher 15 y sin( 60deg) 16 y 4 sin 17 y 3 cos 18 y cos 1 _ 2 19 y 3 cos 2 20 y 4 cos( 60deg)

y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l

21 Draw the uv-coordinate system In each quadrant put a sign or a sign to show whether cos is positive or negative when angle terminates in that quadrant

22 Draw the uv-coordinate system In each quadrant put a sign or a sign to show whether sin is positive or negative when angle terminates in that quadrant

23 Functions of Reference Angles Problem is property relates the sine and cosine of an angle to the sine and cosine of the reference angle Give numerical examples to show that the property is true for both sine and cosine

PROPERTY Sine and Cosine of a Reference Angle

sin ref sin and cos ref cos

24 Construction Problem For this problem use pencil and paper or a computer graphing program such as e Geometerrsquos Sketchpad Construct a right triangle with one horizontal leg with length 8 cm and an acute angle of measure 35deg with its vertex at one end of the 8-cm leg Measure the hypotenuse and the other leg Use these measurements to calculate the values of sin 35deg and cos 35deg from the de nitions of sine and cosine How well do the answers agree with the values you get directly by calculator While keeping the angle measure equal to 35deg increase the lengths of the sides of the right triangle Calculate the values of sin 35deg and cos 35deg in the new triangle What do you nd


Problem Notes (continued)

Problems 7ndash14 aresimilartoExample2andhelpstudentslearnthedefinitionsofsineandcosine

Problems 7ndash14couldbeaccomplishedwiththeunitVcommand

7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

Problems 15ndash20givestudentsachancetoapplytheirtransformationskillsfromChapter1totheparentsineandcosinegraphsAblacklinemasterfortheseproblemsisavailableintheInstructorrsquos Resource Book

ForProblems 15ndash20havestudentsdefinetheparentfunctiononaCASandgraphthetransformationsusingfunctionnotationtomorefullyconnecttransformationsontrigonometricfunctionstotransformationsingeneral

Problems 21 and 22askstudentstomakegeneralizationsaboutwhensinuandcosuarepositiveandwhentheyarenegative

Problem 23presentsthepropertythatrelatesthesineandcosineofanangletothesineandcosineofthereferenceangle

Problem 24requiresstudentstoconstructarighttrianglewithparticularmeasuresTh entheycanobservethatthesineandcosineofanangledonotchangeifthetriangleisdilatedIfstudentsdonotuseacomputergraphingprogramtheywillneedaprotractorforthisproblemCentimetergraphpaperfromtheBlacklineMasterssectionintheInstructorrsquos Resource Bookmaybeused

Seepages999ndash1000foranswerstoProblems10ndash24andCASProblem1

Additional CAS Problems

1 Giventhreeangleswhoseterminalsidescontain(35)(4575)and(126k)respectivelyallhavethesamesineandcosinevaluesa Whatiskb Whatistherelationshipamongall

pointsthathavethesamesineandcosinevaluesasthegiventhreepoints

c Findcoordinatesofthepointwithay-coordinateof100whosesineandcosinevaluesarethesameasthegivenpoints

d Th epoint(ab)is12unitsfromtheoriginbutitssineandcosinevaluesareidenticaltothegivenpointsWhatarethevaluesofaandb

e Th epoint(cc13)hasthesamesineandcosinevaluesasthegivenpointsFindc





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261

S e c t i o n 5 - 4 PL AN N I N G

Class time2days

Homework AssignmentDay 1RAQ1ndashQ10Problems1(4)67

911ndash14Day 2Problems15ndash1921ndash31odd3539

40ndash46

Teaching ResourcesExploration5-4ValuesoftheSineand

CosineFunctionsExploration5-4aValuesoftheSix

TrigonometricFunctionsExploration5-4bDirectMeasurementof

FunctionValuesTest12Sections5-1to5-4FormsA

andBSupplementaryProblems


SineWaveTracer

ActivityTrigonometryTracers

CASActivity5-4aPythagoreanRelationships

TE ACH I N G

Important Terms and ConceptsTangentCotangentSecantCosecantReciprocalpropertiesUnitcircleComplementaryanglesEllipsisformat

Exploration NotesExploration 5-4 requiresstudentstofindvaluesofsineandcosineTh isisagoodactivitytouseatthebeginningofthesection

Seepage264fornotesonadditionalexplorations

1 sinu5v __r cosu5u __r 2 sin123508386cos1235205446

Becauserisalwayspositivesin123ispositivebecausevispositiveinQuadrantIIandcos123isnegativebecauseuisnegativeforpointsinQuadrantII3 r5

____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58

4 Chooseforexampleapointwithr52Bythepropertiesof30-60-90trianglesu51andv52

__3

Sosin30051_2andcos3005 __3___2

Becauserisalwayspositivesin300isnegativebecausevisnegativeinQuadrantIVandcos300ispositvebecauseuispositiveinQuadrantIV5 Answersmayvary

Section 5-4 Values of the Six Trigonometric Functions


Note e coordinates u and v are also the horizontal and vertical displacements of the point (u v) in the reference triangle

The Names Tangent and SecantTo see why the names tangent and secant are used look at Figure 5-4a

e point (u v) has been chosen on the terminal side of angle where r 1 unit e circle traced by (u v) is called the unit circle (a circle with radius 1 unit) e value of the sine is given by

sin v __ r v __ 1 v

us the sine of an angle is equal to the vertical coordinate of a point on the unit circle Similarly cos u _ 1 u the horizontal coordinate of a point on the unit circle

If you draw a vertical line tangent to the circle at the point (1 0) another reference triangle is formed with the adjacent displacement equal to 1 unit In this larger triangle

tan opposite

_______ adjacent length of tangent segment

_____________________ 1 length of tangent segment

Hence the name tangent is used

v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a

DEFINITIONS The Six Trigonometric FunctionsLet (u v) be a point r units from the origin on the terminal side of a rotating ray If is the angle to the ray in standard position then the following denitions hold

Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite

__________ hypotenuse sin vertical coordinate ________________ radius v __ r

cos adjacent

__________ hypotenuse cos horizontal coordinate __________________ radius u __ r

tan opposite

________ adjacent tan vertical coordinate __________________ horizontal coordinate v __ u

cot adjacent

_______ opposite cot horizontal coordinate __________________ vertical coordinate u __ v

sec hypotenuse

__________ adjacent sec radius __________________ horizontal coordinate r __ u

csc hypotenuse

__________ opposite csc radius ________________ vertical coordinate r __ v


Four other ratios can be made using u v and r eir names are tangent cotangent secant and cosecant and their de nitions are given in the box on the next page

e right triangle de nition of tangent for an acute angle is extended to the ratio of v to u for a point on the terminal side of any angle

tan v __ u

vertical displacement ____________________ horizontal displacement

opposite _______ adjacent

e cotangent secant and cosecant functions are reciprocals of the tangent cosine and sine functions respectively e relationship between each pair of functions such as cotangent and tangent is called the reciprocal property of trigonometric functions which you will explore fully in Chapter 7

When you write the functions in a column in the order sin cos tan cot sec and csc the functions and their reciprocals have this pattern

sin cos tan cot sec csc

Reciprocals

cot = 1 ____ tan

3 is gure shows an angle in standard position e terminal side contains the point ( 3 7) Write sin and cos exactly using fractions and radicals

u

v

( 3 7)

4 is gure shows an angle of 300deg in standard position Choose a convenient point on the terminal side Use properties of special

triangles from geometry to nd u v and r for the point you picked en write sin 300deg and cos 300deg exactly using radicals if necessary Explain why sin 300deg is negative but cos 300deg is positive

u

v

300deg


EXPLORATION continued


Section Notes

Section5-4definesthefourremainingtrigonometricfunctionsmdashtangentcotangentsecantandcosecantmdashandintroducesthereciprocalproperties

IfyougavestudentsaTrigonometricRatiosTableinSection5-3havethemfillinmoreofthevaluesasyouintroducetheotherfourratiosYoumightaskthemtoapplyrelationshipsbetweenfunctionssuchascscu5 1____sinu tanu5sinu ____cosu andsinu 5cos(902u)tofilloutthetable

Th issectionwilltakeabouttwodaystocompleteOnthefirstdaycoverthematerialthroughExample2onpage264YoumightbeginthedaybyassigningExploration 5-4suggestingthatstudentsreadtheirbookstofindtheinformationneededtoanswerthequestionsWhenstudentshavefinisheddiscusstheirresultsIfstudentsseemtounderstandthematerialyoumaybeabletoskipsomeoftheexamplesinthetext

Cosine(orco-sine)ismerelyacontractionofcomplement-sineahistoricalnamingreflectingthiscomplementaryrelationshipTh isrelationshipholdstrueforallcofunctionpairsoftrigonometricfunctionsWatchforstudentswhothinkthatafunctionanditscofunctionmdashforexamplesecantandcosecantmdasharereciprocalsEmphasizethecorrectreciprocalrelationshipsandbesuretodiscussProblem43 inclasstoreinforcethatcofunctionsarenotreciprocalfunctions

WhendrawingreferencetrianglestofindthevaluesoftrigonometricfunctionsstudentsoftenmakethemistakeoflabelingallsideswithpositivevaluesEmphasizetheimportanceofthinkingaboutthesignusedforeachlabel

Becausethelabelsforthehorizontalandverticalsidescorrespondtotheu-andv-valuestheymaybepositiveornegativevaluesdependingonthequadrantthe

terminalsideoftheangleisinBecausethevalueofristheradiusofacirclethehypotenuselabelshouldalwaysbepositive

InExample1onpage264notethatsomecalculatorswillalsoevaluatesecantcosecantandtangentdirectly

WhenaradicaloccursinthedenominatorsuchasinExample2thetextdoesnotsuggestthatstudentsrationalizetheexpressionItwasoncecustomaryto

transformfractionscontainingradicalssothatthedenominatorswererationalnumbersTh etechniquehadadvantagesforfindingdecimalapproximationsbecauseitismucheasiertodividebyarationalnumberwhenusinglongdivision





sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263

Question Inwhatquadrantdoesanangleof2250terminate

Answer QuadrantII

Question Describehowsinucosuandtanuvaryasuincreasesfrom0to90

Answer sinuincreasescosudecreasestanuincreases

Nowthatcalculatorsarewidelyavailabledenominatorsusuallyarenotrationalized

Herearesomegoodclassdiscussionquestions

Question TellwhichtrigonometricfunctionsarepositiveinQuadrantsIIIIIIandIV

Answer QuadIallare1QuadIIsinandcscare1QuadIIItanandcotare1QuadIVcosandsecare1

Question Foranglesthatterminateonaquadrantboundarythevaluesofthetrigonometricfunctionsare061orundefinedFindtheexactvaluesofthesixtrigonometricfunctionsof270

Answer sin270521cos27050tan270isundefinedcot27050sec270isundefinedcsc270521



triangle

r __________

( 5) 2 2 2 ___

29

sin vertical _______ radius 2 _____

___ 29

cos horizontal _________ radius 5 _____

___ 29 5 _____

___

29

tan vertical _________ horizontal 2 ___ 5 2 __ 5

cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2

Note You can use the proportions of the side lengths in the 30degndash60degndash90deg triangle and the 45degndash45degndash90deg triangle to nd exact function values for angles whose reference angle is a multiple of 30deg or 45deg Figure 5-4c shows these proportions

radic___

Figure 5-4c

Find exact values (no decimals) of the six trigonometric functions of 300deg

Sketch an angle terminating in Quadrant IV and a reference triangle with u 1 (Figure 5-4d)

sin __

3 _____ 2 __

3 ____ 2 Use the negative square root because v is negative

cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____

__ 3 Use the reciprocal relationship

sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3

To avoid errors in placing the 1 2 and __

3 on the reference triangle remember that the hypotenuse is the longest side of a right triangle and that 2 is greater than

__ 3

Find EXAMPLE 3

Sketch an angle terminating in Quadrant IV and a reference triangle with (Figure 5-4d)

SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b


e hypotenuse of this larger reference triangle is part of a secant line a line that cuts through the circle In the larger triangle

sec hypotenuse

__________ adjacent length of secant segment

____________________ 1 length of secant segment

Hence the name secant is used By the properties of similar triangles for any point (u v) on the terminal side

tan v __ u and sec r __ u

You have seen why the names tangent and secant are used for these ratios You will explore the reason for the pre x co- in the names cosine cotangent and cosecant when you do Problem 43 in the problem set for this section Section 8-3 explains in detail how the reciprocal functions are related to complementary angles

Approximate Values by CalculatorExample 1 shows how you can nd approximate values of all six trigonometric functions by calculator

Evaluate by calculator the six trigonometric functions of 586deg Round to four decimal places

You can nd sine cosine and tangent directly by calculator

sin 586deg 08535507 08536 Note preferred usage of the ellipsis and the sign

cos 586deg 05210096 05210

tan 586deg 16382629 16383

e other three functions are the reciprocals of the sine cosine and tangent functions Notice that the reciprocals follow the pattern described earlier

cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716

Exact Values by GeometryIf you know a point on the terminal side of an angle you can calculate the values of the trigonometric functions exactly Example 2 shows you the steps

e terminal side of angle contains the point ( 5 2) Find exact values of the six trigonometric functions of Use radicals if necessary but no decimals

5 2) in this instance and draw a perpendicular to the horizontal axis


EXAMPLE 1


sin 586deg

SOLUTION

e terminal side of angle trigonometric functions of

EXAMPLE 2

SOLUTION


Differentiating Instructionbull Youwillprobablyneedtopointout

thatinternalellipsesalwaysindicateapatternwhileendellipsesmayormaynotindicateapatternStudentswillhavetodecidewhichbasedoncontextInthisandothersectionsstudentswillseeellipsesattheendofdecimalnumbersindicatingthatthedecimalcontinuesthoughwithoutanyparticularpattern

bull Askstudentshowtheypronouncetangent cotangent secant andcosecant Theymaypronouncethesewordsdifferentlythanyoudo

bull Notethatcotangentissometimesabbreviated ctn

bull Becausetheymayhaveapproachedtrigonometricfunctionsotherthanthroughrighttrianglessomestudentswillneedahandoutdefiningthetrigonometricfunctionsintermsofarighttriangleandrelatingthesedefinitionstoanillustration

bull Makesurestudentsunderstandthedifferencebetweenand5

bull Makesurestudentsunderstandthatsin2u5(sinu)2sinu2

bull ELLstudentsmayneedhelpwiththemeaningofcertainbeforedoingtheReadingAnalysisItisprobablynotfoundintheirbilingualdictionaries

bull IfyouassignProblem43explainthatthepronunciationofcodiffersifitisasyllablebyitselforifithasaconsonantfollowingitinthesyllablemdashhencecofunction andcomplementary

Additional Exploration Notes

Exploration 5-4aexpandsExploration5-4toincludethedefinitionsofallsixtrigonometricfunctionsYoumightuseExploration5-4inclassandassignthisexplorationashomeworkAllow15ndash20minutes

Exploration 5-4breinforcesstudentsrsquounderstandingofthedefinitionsofsine cosine andtangentStudentsmeasure

lengthstofindthetrigonometricvaluesofanglesThisexplorationshouldtakeabout10minutesandcouldbeusedasaquizorincludedaspartofahomeworkassignment

Technology Notes

Problem 45SineWaveTracerProjectasksstudentstoexplorepropertiesofthesinewaveandwritedownwhattheylearnedThiscanbedonewith

thehelpoftheDynamicPrecalculusExplorationofthesamenameatwwwkeymathcomprecalc

ActivityTrigonometryTracersin

theInstructorrsquos Resource BookhasstudentscomputethesinescosinesandtangentsofseveralanglesbasedonthecoordinatesofapointontheunitcircleTheactivityisbasedonradianmeasureofanglesbutitcanbe


triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265

Additional Class Examples1 Findthesixtrigonometricfunctions

ofuifuterminatesinQuadrantIIIandsinu522_3SolutionSketchanangleterminatinginQuadrantIIIasshowninthefigurePickaconvenientpointontheterminalsideBecause

sine5verticalcoordinate________________radius

themostconvenientpointis3unitsfromtheoriginwithaverticalcoordinateof22Th ehorizontalcoordinatecanbefoundbyapplyingthePythagoreantheorem

2 __________

322(22)252 __5

ChoosethenegativesquarerootbecausethehorizontalcoordinateisnegativeforapointinQuadrantIII

2 3

5 u

v

OncethefigureissketchedtheproblembecomesanldquooldrdquoproblemasinExample2

sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5

NotethattanuispositiveifuterminatesinQuadrantIII

cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2

adaptedtousedegreesYoumaywishtosavethisactivityforSection6-5Allow30ndash40minutes

ActivityAdaptableOptionsSeeSection6-5forradian-basedactivitiesthatcouldbeadaptedforusehere

CAS Activity 5-4a PythagoreanRelationships intheInstructorrsquos Resource BookintroducesstudentstoPythagoreanrelationshipsbetweentrigonometricfunctionsStudentsfirstfindthelengthofthesidesofarighttriangleandthenusethesesidestoderivedifferentPythagoreanrelationshipsAllow20ndash25minutes



Q7 Write an equation for the sinusoid in Figure 5-4h

y

1

720deg

Figure 5-4h

Q8 How was the parent cosine function transformed to get the sinusoid in Figure 5-4h

Q9 Sketch the graph of y x 2 Q10 A one-to-one function is

A Always increasing B Always decreasing C Always positive D Always negative E Always invertible

For Problems 1ndash6 nd a decimal approximation of the given function value Round the answer to four decimal places 1 cot 38deg 2 cot 140deg 3 sec 238deg 4 sec( 53deg) 5 csc( 179deg) 6 csc 180deg (Surprising)

For Problems 7ndash10 nd the exact values (no decimals) of the six trigonometric functions of an angle in standard position whose terminal side contains the given point 7 (4 3) 8 ( 12 5) 9 ( 5 7) 10 (2 3)

For Problems 11ndash14 if angle terminates in the given quadrant and has the given function value nd the exact values (no decimals) of the six trigonometric functions of

11 Quadrant II sin 4 __ 5

12 Quadrant III cos 1 __ 3

13 Quadrant IV sec 4

14 Quadrant I csc 13 ___ 12

For Problems 15ndash20 nd the exact values of the six trigonometric functions of the given angle 15 60deg 16 135deg 17 315deg 18 330deg 19 180deg 20 270deg

For Problems 21ndash32 nd the exact value (no decimals) of the given function Try to do this quickly from memory or by visualizing the gure in your head 21 sin 180deg 22 sin 225deg 23 cos 240deg 24 cos 120deg 25 tan 315deg 26 tan 270deg 27 cot 0deg 28 cot 300deg 29 sec 150deg 30 sec 0deg 31 csc 45deg 32 csc 330deg 33 Find all values of from 0deg through 360deg

for which a sin 0 b cos 0 c tan 0 d cot 0 e sec 0 f csc 0

34 Find all values of from 0deg through 360degfor which

a sin 1 b cos 1 c tan 1 d cot 1 e sec 1 f csc 1


Example 4 shows how to nd the function values for an angle that terminates on a quadrant boundary

Without using a calculator evaluate the six trigonometric functions for an angle of 180deg

Figure 5-4e shows an angle of 180deg in standard position e terminal side falls on the negative side of the horizontal axis Pick any point on the terminal side such as ( 3 0) Note that although the u-coordinate of the point is negative the distance r from the origin to the point is positive because it is the radius of a circle e vertical coordinate v is 0

sin 180deg vertical _______ radius 0 __ 3 0 Use the ldquovertical radiusrdquo rather than ldquoopposite hypotenuserdquo

cos 180deg horizontal _________ radius 3 ___ 3 1

tan 180deg vertical _________ horizontal 0 ___ 3 0

cot 180deg 1 ________ tan 180deg 1 __ 0 No value Unde ned because of division by zero

sec 180deg 1 _______ cos 180deg 1 ___ 1 1

csc 180deg 1 _______ sin 180deg 1 __ 0 No value


EXAMPLE 4

Figure 5-4e shows an angle of 180deg in standard position e terminal side falls on the negative side of the horizontal axis Pick any point on the terminal side such

SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis

From what you have read in this section what do you consider to be the main idea In what way does the reference triangle help you explain why certain trigonometric functions have negative values in certain quadrants of a uv-coordinate system How do geometric properties of right triangles allow you to nd the exact values of trigonometric functions of certain angles

Quick Review

Problems Q1ndashQ5 concern the right triangle in Figure 5-4f

ed

f Figure 5-4f

Q1 Which side is the leg opposite angle Q2 Which side is the leg adjacent to angle Q3 Which side is the hypotenuse Q4 cos Q5 sin Q6 Write an equation for the sinusoid in

Figure 5-4g

y

1

720deg

Figure 5-4g

5min

Reading Analysis Q1 Which side is the leg opposite angle

Problem Set 5-4


Additional Class Examples (continued)

2 Findexactvalues(nodecimals)ofthesixtrigonometricfunctionsof45SolutionTh efigureshowsa45angleinstandardpositionAnyperpendicularfromtheterminalsidetothehorizontalaxisformsa45-45-90righttriangleTh usitslegshaveequallengthAconvenientpointtopickis(u v)5(11)FromthePythagoreantheoremthehypotenuselengthisr5

_______121125

__2

1

1

(1 1)

2

45deg u

v

Th evaluesofthesixfunctionscannowbewrittenbyinspectionasbefore

sinu5 1____

__2

cosu5 1____

__2

tanu51__151

cotu 5 1_____tanu 51

secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES

Supplementaryproblemsforthissectionareavailableatwwwkeypresscomkeyonline

Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E

Problems 1ndash6 aresimilartoExample1

Problems 1ndash6aretrivialwhenusingaTI-Nspiregrapher1 12799 2 2119183 218871 4 166165 2572987 6 Undefined

Problems 7ndash10aresimilartoExample27 sinu523__5cosu54__5tanu523__4

cotu524__3secu55__4cscu525__3

y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4


Problems 15ndash32aresimilartoExample3Th eyrequirestudentstouserelationshipsin30-60-90and45-45-90trianglestofindthefunctionvaluesforangleswhosereferenceanglesaremultiplesof30or45

Problems 15ndash32aretrivialwhenusingaTI-Nspiregrapher21 0 22 2 1____

__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22

Problems 33 and 34askstudentstofindalltheanglesforwhichthefunctionvaluesare0or1

Problems 33 and 34canbeevaluatedusingtheSolvecommandwithconstraintsontheangleasshowninthefigurebelowGiventheperiodicityofthetrigonometricfunctionsusingtheSolvecommandwithouttheconstraintsleadstounexpectedanswersNoticethebolditalicizedn10intheanswerAbolditalicizedn(whereisapositiveintegerandindicatesonlythepositionofthisrandomnumberinthegrapherrsquosmemory)istheCASnotationforarandomintegerTh efirstlineofthefigureshowsthesetofallanswerstosinu50is180kwherekisanyintegerTh especialcasesofthissolutionsetfor0u360areshowninline2

8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74

tanu57__5cotu55__7secu52 ___

74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3

Problems 11ndash14askstudentstousethevalueofonetrigonometricfunctionandtheterminatingquadrantoftheangletodeterminethevaluesoftheothertrigonometricfunctionsTh eseproblemsaresimilartoAdditionalClassExample1

11 sinu54__5cosu523__5tanu524__3


Seepages1000ndash1001foranswerstoProblems12ndash2033and34

Inverse Trigonometric Functions and Triangle ProblemsYou have learned how to evaluate trigonometric functions for speci c angle measures Next yoursquoll learn to use function values to nd the angle measures Yoursquoll also learn how to nd unknown side and angle measures in a right triangle

Given two sides of a right triangle or a side and an acute angle nd measures of the other sides and angles

Inverses of Trigonometric FunctionsIn order to nd the measure of an angle when its function value is given you could press the appropriate inverse function keys on your calculator

e symbol cos 1 is the familiar inverse function terminology of Chapter 1 read as ldquoinverse cosinerdquo Note that it does not mean the 1 power of cos which is the reciprocal of cosine

(cos x) 1 1 _____ cos x

Trigonometric functions are periodic so they are not one-to-one functions ere are many angles whose cosine is 08 (Figure 5-5a) However for each trigonometric function there is a principal branch of the function that is one-to-one and includes angles between 0deg and 90deg e calculator is programmed to give the one angle on the principal branch e symbol cos 1 08 means the one angle on the principal branch whose cosine is 08 e inverse of the cosine function on the principal branch is a function denoted cos 1 x

yy 08 Many angles whose cosine is 08

Principal branch is a one-to-one function Figure 5-5a

In Section 7-6 you will learn more about the principal branches of all six trigonometric functions For the triangle problems of this section all the angles will be acute so the value the calculator gives you is the value of the angle you want

Inverse Trigonometric Functions and Triangle Problems

5 -5


Objective

269Section 5-5 Inverse Trigonometric Functions and Triangle Problems268 Chapter 5 Periodic Functions and Right Triangle Problems

For Problems 35ndash42 nd the exact value (no decimals) of the given expression Note that the expression sin 2 means (sin ) 2 You may check your answers by calculator 35 sin 30deg cos 60deg 36 tan 120deg cot( 30deg) 37 sec 2 45deg 38 cot 2 30deg 39 sin 240deg csc 240deg 40 cos 120deg sec 120deg 41 tan 2 60deg sec 2 60deg 42 cos 2 210deg sin 2 210deg 43 Recall that complementary angles sum to 90deg

a If 23deg what is the complement of b Find cos 23deg and nd sin (complement of 23deg)

What relationship do you notice c Based on what yoursquove discovered what do you

think the pre x co- stands for in the names cosine cotangent and cosecant

44 Pattern in Sine Values Problem Find the exact values of sin 0deg sin 30deg sin 45deg sin 60deg and sin 90deg Make all the denominators equal to 2 and all the numerators radicals Describe the pattern you see

45 Sine Wave Tracer Project Figure 5-4i shows the unit circle in a uv-coordinate system and a sinusoid whose y-values equal the v-values of the point P where the rotating ray cuts the unit circle Create this sketch using dynamic geometry so ware such as e Geometerrsquos Sketchpad so that when you move point P around the circle point Q traces the sinusoid You can also nd the Sine Wave Tracer exploration at wwwkeymathcomprecalc that provides this sketch

Explore this sketch Write a paragraph telling what you learned In particular explain the relationship between plotted as an angle in a uv-coordinate system and plotted horizontally in a y-coordinate system

46 Journal Problem Update your journal writing about concepts you have learned since the last entry and concepts about which you are still unsure

360deg270deg180deg90deg0deg 450deg

P (rotate)

Trace

Qv

u

y

Figure 5-4i



Problems 35ndash42introducestudentstothenotationsin2uInitiallywritesin2uas(sinu)2tohelpthemunderstandthisnotation

Problems 35ndash42aretrivialonaTI-Nspiregrapher

35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1

Problem 43helpsstudentsmakesenseoftheprefixco-inthecofunctiontrigonometricfunctionsIntheclassdiscussionofthisproblemaskstudentstocompleteequationssuchasthese

sin165cos_____Answer 74

tan435cot_____Answer 47

sec365_____54Answer csc

43a 67

43b cos235093205sin67509205Th eyareequal43c ldquoComplementrdquo

Problem 44 asksstudentstolookforpatternsintheexactvaluesofsin0sin30sin45sin60andsin90Somestudentsfindthispatterninterestingandrememberitforuseinfutureproblems

44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2

Patterndescriptionswillvary

Problem 45isagoodexercisetobuildunderstandingofthesinefunctionAsthepointProtatesaroundthecirclestudentscanseehowthesinefunctionrelatestotheunitcircle

Seepage1001foranswerstoCASProblems1and2

45 Intheuv-coordinateplaneuistheanglefromtheorigintothepoint(u v)instandardpositionIntheuy-coordinateplaneurepresentsthesameanglebutisnowusedastheindependentvariable

46 Journalentrieswillvary


1 UseyourCASinExactorAutomodetocreateatableofvaluesforsinu andcosu

ifu 50deg10deg20deg30deg40deg50deg60deg70deg80deg90deg

2 WriteanequationthatdescribestherelationshipbetweenthevaluesinthetableinProblem1Useyourknowledgeofrighttriangletrigonometrytoexplainwhyyourequationmustbecorrect





(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day

Homework AssignmentRA Q1ndashQ10Problems1ndash57(8)9ndash11

1314(21)24

SuggestionAssigntheseproblemsonthedaythesectioniscoveredinclassTh enproceedtoChapter6andassignoneadditionalproblemfromSection5-5eachdayforaweekormore

Teaching ResourcesExploration5-5aMeasurementofRight

TrianglesExploration5-5bAccurateRight

TrianglePracticeExploration5-5cEmpireStateBuilding

ProblemTest13Section5-5FormsAandBSupplementaryProblems

TE ACH I N G

Important Terms and ConceptsInversetrigonometricfunctionsPrincipalbranchMathematical-worldanswerReal-worldanswer

Section Notes

Section5-5openswiththedefinitionoftheinverse cosine functionasldquoananglewhosecosineisxrdquoStudentsquicklypickuptheideaofaninversefunctionldquoundoingrdquotheoriginalfunctionItisextremelyimportanttoemphasizethatthenotationcos21xisnotthesameas(cosx )21thereciprocalofcosxTh isisparticularlyconfusingbecauseintheprevioussectiontheexponent2insin2uwasldquomovedrdquotogettheequivalent

expression(sinu)2Explaintostudentsthatthe21incos21xisnotanexponentratheritisashorthandnotationusedtoindicateaninversefunctionRemindthemthattheysawthisnotationinChapter1whenf 21(x)wasusedtorepresenttheinverseoff(x)

SomecalculatorsavoidthisdifficultybyusingthesymbolacosxwhereldquoardquocanbethoughtofasstandingforldquoanglerdquoYoucancallcosndash1xldquoananglewhosecosineisxrdquo

Section 5-5 Inverse Trigonometric Functions and Triangle Problems

Sketch an appropriate right triangle as in Figure 5-5c Label the known and unknown information

h ____ 473 tan 53deg Write a ratio for tangent

h 473 tan 53deg 627692 Solve for h

e tower is about 628 m high Write the real-world answer

Note that the angle must always have the degree sign even during the computations e symbol tan 53 has a di erent meaning as you will learn when you study radians in the next chapter

A ship is passing through the Strait of Gibraltar along a straight path At its closest point of approach Gibraltar radar determines that the ship is 2400 m away Later the radar determines that the ship is 2650 m away (Figure 5-5d)

a By what angle did the shiprsquos bearing from Gibraltar change

b How far did the ship travel between the two observations

2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d

a Draw the right triangle and label the unknown angle By the de nition of cosine

cos adjacent

__________ hypotenuse 2400 ____ 2650

cos 1 2400 ____ 2650 250876deg Take the inverse cosine to nd

e angle measure is about 2509deg

b Label the unknown side d for distance By the de nition of sine

d ____ 2650 sin 250876deg Use the unrounded angle measure that is in your calculator

d 2650 sin 250876deg 11236102

e ship traveled about 1124 m


SOLUTION

473 m

h

53deg

Figure 5-5c

A ship is passing through the Strait of Gibraltar along a straight path At its closest point of approach Gibraltar radar determines that the ship is 2400 m away Later

EXAMPLE 2

a Draw the right triangle and label the unknown angle of cosine

SOLUTION

271Section 5-5 Inverse Trigonometric Functions and Triangle Problems

e de nitions of the inverse trigonometric functions are given in the box

DEFINITIONS Inverse Trigonometric FunctionsIf x is the value of a trigonometric function at angle then the inverse trigonometric functions can be de ned on limited domains

sin 1 x means sin x and 90deg 90deg

cos 1 x means cos x and 0deg 180deg

tan 1 x means tan x and 90deg 90deg

Notes

Words ldquoAngle is the angle on the principal branch whose sine (and so on) is xrdquo

Pronunciation ldquoInverse sine of xrdquo and so on never ldquosine to the negative 1rdquo

e symbols sin 1 x cos 1 x and tan 1 x are used only for the value the calculator gives you not for other angles that have the same function value e symbols cot 1 x sec 1 x and csc 1 x are similarly de ned

e symbol sin 1 x does not mean the reciprocal of sin x

Right Triangle ProblemsTrigonometric functions and inverse trigonometric functions o en come up in applications such as right triangle problems

Suppose you have the job of measuring the height of the local water tower Climbing makes you dizzy so you decide to do the whole job at ground level You nd that from a point 473 m from the base of the tower you must aim a laser pointer at an angle of 53deg (angle of elevation) to hit the top of the tower (Figure 5-5b) How high is the tower

Angle of elevationHow high

Watertower

53

473 m Figure 5-5b

Suppose you have the job of measuring the height of the local water tower Climbing makes you dizzy so you decide to do the whole job at ground level

EXAMPLE 1




Atthispointthetextavoidstheincorrectcalculatoranswerforcot21(negative argument)bynotaskingquestionsofthistypeTh istopiciscoveredinSection7-6wherepropertiesofinversecircularfunctionsarediscussedForcalculatorsthatdonothaveacot keythetankeyisusedtocalculatecot21xTh isiswhyfornegativeargumentscot21xwillincorrectlygiveananswerinQuadrantIVinsteadofinQuadrantIITh erangeoftan21xis[29090]whereastherangeofcot21xis[0180]

StudentswhouseTI-Nspiregrapherscanevaluateallsixinversetrigonometricfunctionsdirectly

Th eprimaryfocusofthesectionistosolveforanglesandorsidesofrighttrianglesinreal-worldcontextsNotethedifferenceinthetextbetweenmathematical-world answerswhichareunroundedandreal-world answers whichareroundedNotethatthesolutionstotheexamplesendwithasummarystatement(egldquoTh etowerisabout628mhighrdquo)WritingasummarystatementafterthemathematicalequationsprovidesstudentsanopportunitytodemonstratethattheyunderstandwhattheirmathematicalworkrepresentsStudentsareaskedtowritesimilarstatementsontheAPCalculusexam

Example1involvesanangle of elevationSeveralhomeworkproblemsinvolve angles of depressionYoumayneedtorefreshyourstudentsrsquomemoriesaboutwhatthesetermsmeanAnangleofelevationliesabovethehorizontalwhereasanangleofdepressionliesbelowthehorizontalTh isillustrationshowsthatanangleofelevationcorrespondstoacongruentangleofdepression

Angle of elevation

Angle of depression

InExample2thephraseldquoatitsclosestpointofapproachrdquoensuresthatarighttriangleisformedStudentsshouldrecallfromgeometrythattheclosestdistancefromapointtoalineistheperpendiculardistancePartbcanalsobesolvedusingthe

PythagoreantheoremEncouragestudentstouseatrigonometricmethodtosolvetheproblemandthentochecktheiranswerswiththePythagoreantheorem

Exploration Notes

Exploration 5-5a introducestheinversetangentfunctionkeyonthecalculatorStudentsneedacentimeterrulerandprotractorforthisactivityAllow15ndash20minutestocompletetheexploration









2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271

Additional Class Example

Th isexampleillustrateshowtofindinversetrigonometricfunctionvaluesforafunctionthatmaynotappearonthecalculator

1 Findcsc2132Solution

u5csc2132 Choosealettertostandfortheangle

cscu532 Usethedefinitionofinversecosecant

1____sinu 532 GetafunctionthatisonthecalculatorCosecantisthereciprocalofsine

sinu5 1___32 Takethereciprocalofbothsidesoftheequation

u 5sin21 1___32Taketheinversesineofbothsides

5182099182

CAS Suggestions

Th eSolvecommandisusefulforreinforcingexactlywhataninversetrigonometricfunctionisdoingSomestudentsprefertouse Solveasanalternativetotheinversefunctionsbecauseitismoreintuitivetothem

Th efirstlineofthefigureshowsthesolutiontoExample1Th elasttwolinesshowthesolutiontoExample2aNoticethatthemiddlelinedidnothaveanyrestrictionsandgaveallpossiblesolutionsfromtheperiodicfunctionHereuseofaCASforcesstudentstoclearlystateandunderstandtheimplicationsofdomainrestrictionsExploration 5-5b emphasizesthatcalculate

meanstofindtheanswerwithequationsusingalgebraandtrigonometricdefinitionsTh isexplorationcanbeusedasa10-minutequiz

Exploration 5-5c workswellasasmall-groupexplorationanexampletoworkwiththewholeclassora15-minutequiz

Diff erentiating Instructionbull ELLstudentsmayneedclarificationon

theworddepressionasitrelatestoanglesBilingualdictionariesmayprovideonlythepsychologicalmeaning

bull Somestudentswillhavelearnedthetermarcfunctionratherthaninverse function


e Berlin Wall long a symbol of the Cold War was demolished in 1989 is man is chiseling away a souvenir piece of the wall

9 Ladder Problem Suppose you have a ladder 27 m long like the one in the picture above

a If the ladder makes an angle of 63deg with the level ground when you lean it against a vertical wall how high up the wall is the top of the ladder

b Your cat is trapped on a tree branch 26 m above the ground If you lean the top of the ladder against the branch what angle does the bottom of the ladder make with the level ground

10 Flagpole Problem You must order a new rope for the agpole To nd out what length of rope is needed you observe that the pole casts a shadow 116 m long on the ground e angle of elevation of the Sun is 36deg at this time of day (Figure 5-5h) How tall is the agpole

Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h

11 e Grapevine Problem Interstate 5 in California enters the San Joaquin Valley through a mountain pass called the Grapevine e road descends from an elevation of 3500 above sea level to 1500 above sea level in a slant distance of 65 mi

a Approximately what angle does the roadway make with the horizontal

b What assumption must you make about how the road slopes

12 Grand Canyon Problem From a point on the North Rim of the Grand Canyon a surveyor measures an angle of depression of 13deg to a point on the South Rim (Figure 5-5i) From an aerial photograph she determines that the horizontal distance between the two points is 10 mi How many feet is the South Rim below the North Rim

10 miNorthRim South

RimAngle ofdepression

13deg

Figure 5-5i

13 Tallest Skyscraper Problem In 2005 Taipei 101 in Taiwan was the worldrsquos tallest skyscraper reaching 509 m above the ground Suppose that at a particular time the building casts a shadow on the ground 1070 m long What is the Sunrsquos angle of elevation at this time


Problem Set 5-5

Reading Analysis

From what you have read in this section what do you consider to be the main idea In what way does the inverse of a trigonometric function compare with the inverse of a function that you studied in Section 1-5 How can you use the inverse of a trigonometric function to nd an unknown angle measure in a right triangle

Quick Review

Problems Q1ndashQ6 refer to the right triangle in Figure 5-5e Q1 sin Q2 cos Q3 tan Q4 cot Q5 sec Q6 csc Q7 sin 60deg (No decimals) Q8 cos 135deg (No decimals) Q9 tan 90deg (No decimals) Q10 e graph of the periodic function y cos is

called a

For Problems 1ndash6 evaluate the inverse trigonometric function for the given value (Note the domain restrictions for inverse trigonometric functions given on page 270) 1 Find sin 1 03 Explain what the answer means 2 Find cos 1 02 Explain what the answer means 3 Find tan 1 7 Explain what the answer means 4 Explain why sin 1 2 is unde ned 5 Find cos( sin 1 08) Explain based on the

Pythagorean theorem why the answer is a rational number

6 Find sin( cos 1 028) Explain based on the Pythagorean theorem why the answer is a rational number

7 Principal Branches of Sine and Cosine Problem Figure 5-5f shows the principal branch of the function y sin as a solid line on the sine graph Figure 5-5g shows the principal branch of the function y cos as a solid line on the cosine graph

y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g

a Why is neither the entire sine function nor the entire cosine function invertible

b What are the domains of the principal branches of the cosine and sine functions What property do these principal branches have that makes them invertible functions

c Find sin 1 ( 09) Explain why the answer is a negative number

8 Construction Problem Draw a right triangle to scale with one leg 8 cm long and the adjacent acute angle 34deg Draw on paper with ruler and protractor or on the computer with a program such as e Geometerrsquos Sketchpad

a Measure the lengths of the opposite leg and the hypotenuse to the nearest 01 cm

b Calculate the lengths of the opposite leg and the hypotenuse using appropriate trigonometric functions Show that your measured values and the calculated values agree within 01 cm

5min

xz

yFigure 5-5e



PRO B LE M N OTES

SupplementaryproblemsforthissectionareavailableatwwwkeypresscomkeyonlineTh eseproblemsaresimilartothesectionproblemsQ1 z __ y Q2 x __ y

Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2

Q9 Undefined Q10 Sinusoid1 174576becausesin1745765032 784630becausecos7846305023 818698becausetan818698574 21sinu1forallu5 cos(sin2108)506u5sin2108representsanangleofarighttrianglewithsides34and56 sin(cos21028)5096u5cos21028representsanangleofarighttrianglewithsides724and257a Th eyarenotone-to-one7b Sine290u90Cosine0u180Th eyareone-to-one7c sin21(209)52641580Ontheprincipalbranchonlynegativeanglescorrespondtonegativevaluesofthesine

Problem 8 givesyourstudentsanopportunitytouseTh eGeometerrsquosSketchpad8a Oppositeleg54cmhypotenuse96cm8b Oppositeleg553960cmhypotenuse596497cm






Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273

Problems 9ndash24 canbesolvedonaCASusingaSolvecommandwithoccasionalrestrictionsonu9a 24m

9b 744

Problems 10ndash14involveanglesofelevationanddepression10 84m

11a 3611b Assumethattheslopeoftheroadisconstant12 1198ft

13 254


19 Highland Drive Problem One of the steeper streets in the United States is the 500 block of Highland Drive on Queen Anne Hill in Seattle To measure the slope of the street Tyline held a builderrsquos level so that one end touched the pavement e pavement was 144 cm below the level at the other end e level itself was 71 cm long (Figure 5-5m)

710 cmLevel

Pavement

144 cm

Figure 5-5m

a What angle does the pavement make with the level

b A map of Seattle shows that the horizontal length of this block of Highland Drive is 365 How much longer than 365 is the slant distance up this hill

c How high does the street rise in this block

20 Planet Diameter Problem You can nd the approximate diameter of a planet by measuring the angle between the lines of sight to the two sides of the planet (Figure 5-5n)

Distance

AngleEarth Otherplanet

Figure 5-5n

a When Venus is closest to Earth (25000000 mi) the angle is 0deg1 25 (0 degrees 1 minute 25 seconds) Find the approximate diameter of Venus

b When Jupiter is closest to Earth (390000000 mi) the angle is 0deg0 469 Find the approximate diameter of Jupiter

c Check an encyclopedia an almanac or the Internet to see how close your answers are to the accepted diameters

21 Window Problem Suppose you want the windows of your house built so that the eaves completely shade them from the sunlight in the summer and the sunlight completely lls the windows in the winter e eaves have an overhang of 3 (Figure 5-5o)

a How far below the eaves should the top of a window be placed for the window to receive full sunlight in midwinter when the Sunrsquos noontime angle of elevation is 25deg

b How far below the eaves should the bottom of a window be placed for the window to receive no sunlight in midsummer when the Sunrsquos angle of elevation is 70deg

c How tall will the windows be if they meet both requirements

deg

deg

Figure 5-5o


14 Airplane Landing Problem Commercial airliners y at an altitude of about 10 km Pilots start descending toward the airport when they are far away so that the airplane will not have to dive at a steep angle

a If the pilot wants the planersquos path to make an angle of 3deg with the ground at what horizontal distance from the airport must she start descending

b If she starts descending when the plane is at a horizontal distance of 300 km from the airport what angle will the planersquos path make with the horizontal

c Sketch the actual path of the plane just before and just aer it touches the ground

15 Radiotherapy Problem A doctor plans to use a beam of gamma rays to treat a tumor that is 57 cm beneath the patientrsquos skin To avoid damaging an organ the radiologist moves the source over 83 cm (Figure 5-5j)

a At what angle to the patientrsquos skin must the radiologist aim the source to hit the tumor

b How far will the beam travel through the patientrsquos body before reaching the tumor

83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j

16 Triangular Block Problem A block bordering Market Street is a right triangle (Figure 5-5k) You take 125 paces on Market Street and 102 paces on Pine Street as you walk around the block

Pine St

Market St

Front St

Figure 5-5k

a At what angle do Pine and Market Streets intersect

b How many paces must you take on Front Street to complete the trip

17 Surveying Problem When surveyors measure land that slopes signicantly the slant distance they measure is longer than the horizontal distance they must draw on the map Suppose that the distance from the top edge of the Cibolo Creek bed to the edge of the water is 378 m (Figure 5-5l) e land slopes downward at an angle of 276deg to the horizontal

a What is the horizontal distance from the top of the creek bed to the edge of the creek

b How far below the level of the surrounding land is the surface of the water in the creek

CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l

18 Submarine Problem As a submarine at the surface of the ocean makes a dive its path makes an angle of 21deg with the surface

a If the submarine travels for 300 m along its downward path how deep will it be What is its horizontal distance from its starting point

b How many meters must it go along its downward path to reach a depth of 1000 m



Problem Notes (continued)14a 191km

14b 1914c

15a 34515b 101cm

16a 35316b 72paces17a 335m17b 175m18a 108mdeep280mfromstartingpoint18b 2790m

Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275

19a 11519b 7ft19c 74ft20a 7575mi20b 88680mi20c Venus7521miJupiter88846mi

Problem 21 focusesonhowaneaveandtheplacementofawindowcanbeusedtoshieldthesuninthesummerandallowlightinthewinterThisisaparticularlyniceproblemthatrequiresstudentstousegeometry21a 1ft5in21b 8ft3in21c 6ft10in


Chapter Review and Test is chapter introduced you to periodic functions You saw how these functions occur in the real world For instance your distance from the ground as you ride a Ferris wheel changes periodically e sine and cosine functions are periodic By dilating and translating these functions you can get sinusoids of di erent proportions with their critical points located at di erent places You also learned about the other four trigonometric functions and their relationship to the sine and cosine functions Finally you learned how to use these six functions and their inverses to nd unknown side lengths and angle measures in right triangles

Chapter Review and Test is chapter introduced you to periodic functions You saw how these functions

5 - 6

R0 Update your journal with what you have learned since the last entry Include things such as

How angles can have measures that are negative or greater than 180deg and what reference angles are e de nitions of sine cosine tangent cotangent secant and cosecantWhy sine and cosine graphs are periodicInverse trigonometric functions used to nd angle measuresApplications to right triangle problems

R1 Hose Reel Problem You unwind a hose by

turning the crank on a hose reel (Figure 5-6a) As you crank the distance your hand is above the ground is a periodic function of the angle through which the reel has rotated (Figure 5-6b solid graph) e distance y is measured in feet and the angle is measured in degrees a e dashed graph in Figure 5-6b is the

pre-image function y sin Plot this

parent sine function graph on your grapher Does the result agree with Figure 5-6b

b e solid graph in Figure 5-6b is a dilation and translation of the pre-image function y sin Write the full names of these transformations and write an equation for the function When you plot the transformed graph on your grapher does the result agree with Figure 5-6b

c What is the name for the periodic graphs in Figure 5-6b

Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b

R0 Update your journal with what you have learned parent sine function graph on your grapher

Review Problems

277Section 5-6 Chapter Review and Test

22 Grand Piano Problem A 28-in prop holds open the lid on a grand piano e base of the prop is 55 in from the lidrsquos hinge

55 in

Hinge

Where

90degLid

Prop 28 in

a What angle does the lid make with the piano top when the prop is placed perpendicular to the lid

b Where on the lid should the prop be placed to make the right angle in part a

c e piano also has a shorter (13-in) prop Where on the lid should this prop be placed to make a right angle with the lid

23 Handicap Ramp Project In this project you will measure the angle a typical handicap access ramp makes with the horizontal

a Based on handicap access ramps you have

seen make a conjecture about the angle these ramps make with the horizontal Each member of your group should make his or her own conjecture

b Find a convenient handicap access ramp With a level and ruler measure the run and the rise of the ramp Use these numbers to calculate the angle the ramp makes with the horizontal

c Check on the Internet or use another reference source to nd the maximum angle a handicap access ramp is allowed to make with the horizontal

24 Pyramid Problem e Great Pyramid of Cheops in Egypt has a square base measuring 230 m on each side e faces of the pyramid make an angle of 51deg50 with the horizontal (Figure 5-5p)

a How tall is the pyramid b What is the shortest distance you would have

to climb to get to the top c Suppose you decide to make a model of the

pyramid by cutting four isosceles triangles out of cardboard and gluing them together How large should you make the base angles of these isosceles triangles

d Show that the ratio of the distance you calculated in part b to one-half the length of the base of the pyramid is very close to the golden ratio

__ 5 1 ______ 2

e Check on the Internet or use another reference source to nd other startling relationships among the dimensions of this pyramid Also visit Public Broadcasting Systemrsquos website and look for Nova Secrets of Lost Empires for photographs and information about the pyramids

51deg50ʹ

230 m

Figure 5-5p




22a 30622b 47infromthehinge22c 53infromthehinge23 Answerswillvary

Problem 24 providesanopportunityforstudentstoworkinthreedimensionsYoumaywishtohavestudentsactuallybuildthemodeldescribedintheproblem24a 146m24b 186m

24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary


1 DetermineanothertrigonometricfunctionthatcouldhavebeenusedinExample2aandfindthesameresultsusingthenewfunction

2 UsetheresultsofProblem1tofindtheanswertoExample2awithoutusinganinversetrigonometricfunction

3 StanleyKuppswasflyinghiskitewhenitgotstuckinthegutterofabuildingHedidnrsquotknowhowtallthebuildingwasbutdidnoticethattheyardsurroundingitwasentirelyflatHestoodagainstthesideofthebuildingwiththekitestringpulledtightandheldagainstthegroundHewalkedoutfromthehouseuntil4feetofstringhadreeledoutandestimatedthattheanglethestringmadewiththebuildingwasnowabout25Abouthowhighabovethegroundarethebuildingrsquosgutters

Seepage1001foranswerstoCASProblems1ndash3



5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277


Class Time2days(including1dayfortesting)

Homework AssignmentDay 1ProblemsR0ndashR5T1ndashT22Day 2(afterChapter5Test)ProblemsC1

andC3or ProblemSet6-1

Teaching ResourcesTest14Chapter5FormsAandBSupplementaryProblems

TE ACH I N G

Important Terms and ConceptsCycleofasinusoidalfunctionSlope

Section Notes

Section5-6containsasetofreviewproblemsasetofconceptproblemsandachaptertestTh ereviewproblemsincludeoneproblemforeachsectioninthechapterYoumaywishtousethechaptertesteitherasanadditionalsetofreviewproblemsorashomeworkforthetest

Encouragestudentstopracticetheno-calculatorproblemswithoutacalculatorsothattheyarepreparedforthetestproblemsforwhichtheycannotuseacalculator

PRO B LE M N OTES


StudentscanapproachProblem R1bbygraphingandtransformingasineorcosineparentgraphasdescribedinSection5-3

R0 JournalentrieswillvaryR1a Th egraphsmatchR1b y-dilationby07y-translationby12y52107sinuTh eresultagreeswiththegraphR1c Sinusoid

Section 5-6 Chapter Review and Test

Chapter TestConcept Problems C1 Tide Problem e average depth of the water at

the beach varies with time due to the motion of the tides Figure 5-6c shows the graph of depth in feet versus time in hours for a particular beach Find the four transformations of the parent cosine graph that would give the sinusoid shown Write an equation for this particular sinusoid assuming 1 degree represents 1 h Con rm your answer on your grapher

y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c

C2 Figure 5-6d shows three cycles of the sinusoid function y 10 sin e horizontal line y 3 cuts each cycle at two points a Estimate graphically the six values of

where the line intersects the sinusoid b Calculate the six points in part a numerically

using the intersect feature of your grapher

c Calculate the six points in part a algebraically using the inverse sine function

360deg

3

y

Figure 5-6d

C3 On your grapher make a table of values of cos 2 sin 2 for each 10 degrees starting at 0deg What is the pattern in the answers Explain why this pattern applies to any value of

C4 A ray from the origin of a uv-coordinate system starts along the positive u-axis and rotates around and around the origin e slope of the ray depends on the angle through which the ray has rotated a Sketch a reasonable graph of the slope as a

function of the angle of rotation b What function on your grapher is the same

as the one you sketched in part a


Chapter Test

Part 1 No calculators allowed (T1ndashT8) T1 Sketch an angle in standard position whose

terminal side contains the point (3 4) Sketch the reference triangle showing the reference angle and the displacements u v and r Find the exact values of the six trigonometric functions of

T2 Sketch a 120deg angle in standard position Sketch the reference triangle Show the reference angle and its measure and the displacements u v and r Find the exact values of the six trigonometric functions of 120deg

T3 Sketch a 225deg angle in standard position Sketch the reference triangle Show the

reference angle and its measure and the displacements u v and r Find the exact values of the six trigonometric functions of 225deg

T4 Sketch a 180deg angle in standard position Pick a point on the terminal side and write the horizontal coordinate vertical coordinate and radius Use these numbers to nd the exact values of the six trigonometric functions of 180deg

T5 e number of hairs on a personrsquos head and his or her age are related Sketch a reasonable graph

T6 e distance between the tip of the ldquosecondrdquo hand on a clock and the oor depends on time Sketch a reasonable graph

Part 1 No calculators allowed (T1ndashT8) reference angle and its measure and the

Chapter Test

R2 For each angle measure sketch an angle in standard position Mark the reference angle and nd its measure a 110deg b 79deg c 2776deg

R3 a Find sin and cos given that the terminal side of angle contains the point ( 5 7)

b Find decimal approximations of sin 160deg and cos 160deg Draw an angle of 160deg in standard position and mark the reference angle Based on displacements in the reference triangle explain why sin 160deg is positive but cos 160deg is negative

c Sketch the graphs of the parent sinusoids y cos and y sin

d In which two quadrants on a uv-coordinate system is sin negative

e For the function y 4 cos 2 what are the transformations of the parent function graph y cos Sketch the graph of the transformed function

R4 a Find a decimal approximation of csc 256deg b Find exact values (no decimals) of the six

trigonometric functions of 150deg c Find the exact value of sec if ref 45deg and

angle terminates in Quadrant III d Find the exact value of cos if the terminal

side of angle contains the point ( 3 5) e Find the exact value of sec( 120deg) f Find the exact value of tan 2 30deg csc 2 30deg g Explain why tan 90deg is undened

R5 a Find a decimal approximation of cos 1 06 What does the answer mean

b Sunken Ship Problem Imagine that you are on a salvage ship in the Gulf of Mexico Your sonar system has located a sunken ship at a slant distance of 683 m from your ship with an angle of depression of 28deg i How deep is the water at the location of

the sunken ship ii How far must your ship go to be directly

above the sunken ship iii Your ship moves horizontally toward the

sunken ship Aer 520 m what is the angle of depression

iv How could the crew of a shing vessel use the techniques you used to solve this problem while searching for schools of sh



Differentiating Instructionbull Gooverthereviewproblemsinclass

perhapsbyhavingstudentspresenttheirsolutionsYoumightassignstudentstowriteuptheirsolutionsbeforeclassstarts

bull BecausemanyculturesrsquonormshighlyvaluehelpingpeersELLstudentsoftenhelpeachotherontestsYoucanlimitthistendencybymakingmultipleversionsofthetest

bull ConsidergivingagrouptestthedaybeforetheindividualtestsothatstudentscanlearnfromeachotherastheyreviewandtheycanidentifywhattheydonrsquotknowpriortotheindividualtestGiveacopyofthetesttoeachgroupmemberhavethemworktogetherthenrandomlychooseonepaperfromthegrouptogradeGradethetestonthespotsostudentsknowwhattheyneedtoreviewfurtherMakethistestworth1_3thevalueoftheindividualtestorless

bull ELLstudentsmayneedmoretimetotakethetest

bull ELLstudentswillbenefitfromhavingaccesstotheirbilingualdictionarieswhiletakingthetest

bull InProblem R5bstudentsmayfindithardtounderstandslantandangle of depressiononahorizontalsurfaceYoumayneedtodrawthefiguregivingavisualrepresentationofthewordsintheproblem

bull ProblemssuchasProblems T5 and T6dependonacompleteunderstandingofEnglishIfyouusesimilarproblemsyoumayneedtohelpstudentsunderstandtheprogressiononehairtwohairsthreehairsmanyhairshair andtheconceptofthetip of the second hand

bull ForProblems T18ndashT20youmayneedtodefinesonar

Problem Notes (continued)R2a uref570 R2b uref579

v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test



















R3d QuadrantsIIIandIVR3e y-translationof14u-dilationof1__2

2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3

R4g tan9051__0whichisundefined

Problem R5bisstraightforwardformoststudentsusingaSolvecommandRemembertoincludeanglerestrictionsforProblem R5biiiR5a 531301thismeansthatcos531301506R5b i 321m R5b ii 603m

R5b iii 75R5b iv Fishingcrewscouldusethistechniquetofindtheslantdistanceanddepthofaschooloffish

Problem C1 preparesstudentsforChapter6Th isworkswellasagroupproblem

StudentscanapproachProblem C1bygraphingandtransformingasineorcosineparentgraphasdescribedinSection5-3

Problem C3 preparesstudentsforChapter7Allstudentsshouldbeabletocompletethisproblemindividually

Problem C4 preparesstudentsforcalculusandwillprobablybeachallengeevenforyourbeststudents

Seepages1001ndash1002foranswerstoProblemsC1ndashC4andT1ndashT6

Problem R3acanbesolvedusingaunitVcommand

R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref

Th eangle160terminateswithinQuadrantIIabovethex-axis(sosin1600)andtotheleftofthey-axis(socos1600)R3c

1

1

y

360deg

y cos

y sin

T7 Only one of the functions in Problems T5 and T6 is periodic Which one is that

T8 Figure 5-6e shows the graph of y sin (dashed) and its principal branch (solid) Explain why the function y sin is not invertible but the function dened by its principal branch is invertible

y

90deg90deg

1

1

Figure 5-6e

Part 2 Graphing calculators allowed (T9ndashT22)For Problems T9ndashT12 use your calculator to nd each value T9 sec 39deg T10 cot 173deg T11 csc 191deg T12 tan 1 09 Explain the meaning of the answer T13 Calculate the length of side x

x

T14 Calculate the length of side y

y4 mi21deg

T15 Calculate the measure of angle B

3 m

28 mB

T16 Calculate the length of side z

67 cm

z18deg

T17 Calculate the measure of angle A

24 cm6 cm

A

Buried Treasure Problem For Problems T18ndashT20 use Figure 5-6f the diagram of a buried treasure From the point on the ground at the le of the gure sonar detects the treasure at a slant distance of 193 m at an angle of 33deg with the horizontal

107 mGround

193 m

Buried treasure

33deg

Figure 5-6f

T18 How far must you go from the point on the le to be directly over the treasure

T19 How deep below the ground is the treasure T20 If you keep going to the right 107 m from the

point directly above the treasure at what angle would you have to dig to reach the buried treasure

T21 In Figure 5-6g the solid graph shows the result of three transformations applied to the parent function y cos (dashed) Write the equation for the transformed function Check your results on your grapher

y

2

2360deg 720deg

Figure 5-6g

T22 What did you learn as a result of taking this test that you didnrsquot know before



Problem Notes (continued)T7 Th efunctioninProblemT6isperiodicT8 Th efunctionwiththedashedgraphisnotone-to-onebutthefunctionwiththesolidgraphisone-to-oneT9 12867T10 281443T11 252408T12 419872thismeanstan419872509

Problems T13ndashT21areefficientlyhandledwithaSolvecommandusingtheappropriateanglerestrictionsT13 212762ftT14 42845mi

T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m

T20 444909T21 y52412cos 6__5u T22 Answerswillvary

Unit Overview e central focus of this unit is a study of trigonometric and periodic functions In Chapter 5 students are introduced to the sine and cosine functions the six trigonometric function de nitions inverse trigonometric functions and solving right-triangle problems In this chapter the domains of the trigonometric functions are acute angles measured in degrees Radian measure is introduced in Chapter 6 allowing students to expand the possible domain values of the trigonometric functions to be all real numbers thus generating the corresponding circular functions Used in this way the domain might represent time or distance making circular functions particularly useful for studying real-world applications sinusoidal functions are used as mathematical models to make predictions In Chapter 7 students explore trigonometric properties and identities is chapter covers basic trigonometric properties such as the Pythagorean identities Proving identities helps students learn the properties sharpen their algebraic skills and practice writing algebraic proofs Students also learn to use these properties to solve trigonometric equations In Chapter 8 students extend their study of trigonometric properties to some more complicated properties Graphical investigations of sums and products of sinusoids with unequal periods support an algebraic study of these properties In Chapter 9 students learn to solve problems involving oblique triangles using the law of cosines and the law of sines Students also study the ambiguous case of a triangle and are introduced to vectors

Using This Unit e study of periodic functions follows logically from the rst unit on algebraic exponential and logarithmic functions Unit 2 is central to any precalculus course and students preparing to take calculus will need a solid grasp of the concepts If you want to study trigonometry early in the year this unit can be taught following Chapter 1 without impacting the remainder of the courseFor students who have not studied trigonometry before Chapter 5 provides a solid foundation If students have already mastered this material you may wish to do a few review problems and move on to the next chapter

245



555555

y

245

















C h a p t e r 5












75-6 ChapterReview

R0ndashR5T1ndashT22










55-6 ChapterTest





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu



Class Time1__2day



Functions



TE ACH I N G


Section Notes






90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m


245



555555

y

245

















C h a p t e r 5












75-6 ChapterReview

R0ndashR5T1ndashT22










55-6 ChapterTest





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu



Class Time1__2day



Functions



TE ACH I N G


Section Notes






90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m












C h a p t e r 5












75-6 ChapterReview

R0ndashR5T1ndashT22










55-6 ChapterTest





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu



Class Time1__2day



Functions



TE ACH I N G


Section Notes






90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m













75-6 ChapterReview

R0ndashR5T1ndashT22










55-6 ChapterTest





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu



Class Time1__2day



Functions



TE ACH I N G


Section Notes






90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu



Class Time1__2day



Functions



TE ACH I N G


Section Notes






90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





90deg 540degAngle

Hei

ght (

m)

720deg

y

180deg 360deg

2

11

20




5 -1

11 m

Ground

y Height

Seat

AngleRadius9 m

Rotation


Objective


20


y

x

Figure 5-1c






cos u




y cos

0deg 1

30deg 08660

60deg 05

90deg 0


1y



ALGEBRAICALLY

NUMERICALLY

GRAPHICALLY

VERBALLY



v

(u v)

rv

uu


Exploration Notes


Technology Notes


PRO B LE M N OTES






v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a







Class Time1day


272930



Conversions

TE ACH I N G


Section Notes







v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




v

u

133deg

v

u493deg

493deg 133deg 360deg(1)

v

u

227deg





360degn












v

u

133deg(u v)

v

u

251deg

(u v)

v

u560deg

(u v)

Figure 5-2b


5 -2


v

u57deg

(u v)

Fixedpoint

Terminal position

Initial position

Rotatingray

Angle

Figure 5-2a


















v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2






6 uref5u

ref

v

u

7 uref51801(2150)530 v

30deg150deg

u



Exploration Notes


28deg152deg

v

u

2 uref52502180570

250deg

70deg

v

u


5 uref53602310550

50deg310deg

v

u






v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







v

u

ref 71deg

71degref 71deg

v

u 133deg

ref 47deg


u 254deg

ref 74deg


317deg ref 43deg

v

u


Figure 5-2d







EXAMPLE 1


SOLUTION


EXAMPLE 2







v

u152deg


v

u250deg






v

u

150deg2





251

Technology Notes


CAS Suggestions





______2221252

__3(2

__3becauseit


30deg1

12

2

3

v

u

9 Answersmayvary





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min


PRO B LE M N OTES


Q9 200 Q10 A


v

uref

v

uref


u

ref

v

uref


u

ref

v

u

ref

7 uref554 8 uref577

v

uref

v

u

ref

11

2

4y

x


uref

v

uref


uref

v

u

ref




21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





21 145deg37 22 268deg29

23 213deg16 24 121deg43

25 308deg14 51 26 352deg16 44





30

y

f

g

10

x10 10

10

y

f

g

10

x10 10

10


Problem Set 5-2


ref 37deg

c 217deg

v

u 4897deg



v

u

Figure 5-2e




Reading Analysis







Q10 x 20 ___ x 5



5min

253


v

uref

v

u

ref

23 uref53316 24 uref55817v

uref

v

uref

25 uref551459 26 uref574316v

uref

v

u

ref


27



Distance


uref

v

uref

15 uref5259 16 uref5868

v

uref

v

uref







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3




Class Time1day






SineWaveTracer


TE ACH I N G








v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







v(u v)

u

v

u


Draw aperpendicular

Ray rotates

Referencetriangle




v

u

Figure 5-3c



sin opposite leg






y sin

1

1

360deg 720deg

360deg 720deg

1

1

y cos

Figure 5-3a




One period


Figure 5-3b


5 -3


255

Section Notes









y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f














y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m





y 1

1

Decreasing

Decreasing

Increasing

Increasing

Repeating


Figure 5-3g





cos 147deg 08386




v

u






EXAMPLE 1


SOLUTION

v

u33deg147deg

Figure 5-3h



u neg

v pos r

ref

v(u v)

u

in Quadrant II

u neg

v neg

(u v)

refr

v

u

in Quadrant III

u pos

r

v

u

v negref

(u v)



sin v __ r


u __ r






sin v 1 __ r 1

v 2 __ r 2

and cos u 1 __ r 1

u 2 __ r 2

v

u


Figure 5-3e

v

u

Similartriangles

(u1 v1)(u2 v2)

r1

r2

Figure 5-3f

257






Exploration Notes








y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



Technology Notes



CAS Suggestions





y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



y

1360deg


Sketch the graphy

1360deg

Figure 5-3k

Reading Analysis



function











5min


Problem Set 5-3




r __________

8 2 ( 5) 2 ___

89 Show ___

89 on the gure

sin 5 _____



cos 8 _____




y

1

360deg

Figure 5-3j





EXAMPLE 2


SOLUTION

v

u8

5__89

Figure 5-3i


EXAMPLE 3



PRO B LE M N OTES


v

uref


v

uref


v

u

ref

cos2005209396cos20509396cos20052cos20


Q7

Q8 1Q9 120Q10 20

1 uref570

v

uref

sin2505209396sin70509396sin25052sin70

Height

Distance





sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l











7 sinu 5 11______

____170508436

cosu5 7______

____170505368

8 sinu5 1_____

___17502425

cosu5 4_____

___17509701

9 sinu5 5_____

___29509284

cosu522_____

___295203713

















sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






sin v __ r



cos u


____________________ radius adjacent




5 - 4


Objective



u

r

v(u v)


u

r

v

(u v)

123deg

continued




y

1

360degy sin

y

1

360deg

y cos

Figure 5-3l







261


Class time2days



40ndash46







SineWaveTracer



TE ACH I N G






____________ 2321 2725

___58so

sinu527_____

___58andcosu523_____

___

58


__3

Sosin30051_2andcos3005 __3___2







sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg




Section Notes















sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






sin v __ r v __ 1 v



tan opposite




v

(u v)

u

Tangent

Secant

Sine

(1 0)1

1

Figure 5-4a


Right Triangle Form

v

(u v)

Opposite

Hypotenuser

leg v

Adjacent legu

u

Coordinate Form

sin opposite


cos adjacent


tan opposite


cot adjacent


sec hypotenuse


csc hypotenuse





tan v __ u






Reciprocals

cot = 1 ____ tan


u

v

( 3 7)



u

v

300deg



263


Answer QuadrantII











triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION















Technology Notes






triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



triangle

r __________

( 5) 2 2 2 ___

29


___ 29


___ 29 5 _____

___

29


cot 1 _____ tan 5 __ 2

sec 1 _____ cos ___

29 _____ 5

csc 1 ____ sin ___

29 _____ 2


radic___

Figure 5-4c



sin __

3 _____ 2 __


cos 1 __ 2

tan __

3 ____ 1 __

3 Simplify

cot 1 _____ tan 1 ____


sec 1 _____ cos 2 __ 1 2

csc 1 ____ sin 2 ____

__ 3



__ 3

Find EXAMPLE 3


SOLUTION

60deg

300degv

u1

2

(u v)

__3

Figure 5-4d

v

u

( 5 2)

5

2 radic___

29

Figure 5-4b



sec hypotenuse










cos 586deg 05210096 05210

tan 586deg 16382629 16383


cot 586deg 1 ________ tan 586deg 06104026 06104

sec 586deg 1 ________ cos 586deg 19193503 19194

csc 586deg 1 _______ sin 586deg 11715764 11716





EXAMPLE 1


sin 586deg

SOLUTION


EXAMPLE 2

SOLUTION

265





2 __________

322(22)252 __5


2 3

5 u

v


sinu 522___3 522___3 Given

cosu52 __5_____3 52

__5____3

tanu5 22_____2

__55 2____

__5


cotu5 1_____tanu 5 __5____2

secu5 1_____cosu 52 3____

__5

cscu5 1____sinu 523__2







y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




_______121125

__2

1

1

(1 1)

2

45deg u

v


sinu5 1____

__2

cosu5 1____

__2

tanu51__151


secu5 1_____cosu 5 __2

cscu5 1____sinu 5 __2

PRO B LE M N OTES


Q1 d Q2 e

Q3 f Q4 e _ f

Q5 d __ f Q6 y5sinu

Q7 y54cosu

Q8 y-dilationby4

Q9

Q10 E





y

x



y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




y

1

720deg

Figure 5-4h
























sec 180deg 1 _______ cos 180deg 1 ___ 1 1



EXAMPLE 4


SOLUTION

180deg

v

uv 0

u 3r 3

Figure 5-4e

Reading Analysis


Quick Review


ed

f Figure 5-4f


Figure 5-4g

y

1

720deg

Figure 5-4g

5min


Problem Set 5-4




__2

23 21__2 24 21__2

25 21 26 undefined

27 undefined 28 2 1____

__3

29 2 2____

__3 30 1

31 __2 32 22



8 sinu55___13cosu5212___13tanu525___12

cotu5212___5secu5213___12cscu513___5

9 sinu52 7_____

___74cosu52 5_____

___

74


74_____5

cscu52 ___

74_____7

10 sinu5 3_____

___13cosu5 2_____

___

13tanu53__2

cotu52__3secu5 ___

13_____2 cscu5 ___

13_____3


11 sinu54__5cosu523__5tanu524__3







(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i





35 1 36 22 __3

37 2 38 3

39 1 40 141 21 42 1





43a 67



44 sin05 __0____2 sin305

__1____2

sin455 __2____2 sin605

__3____2

sin905 __4____2














(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m






(cos x) 1 1 _____ cos x






5 -5


Objective











P (rotate)

Trace

Qv

u

y

Figure 5-4i

269


Class Time 1day


1314(21)24






TE ACH I N G


Section Notes













2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1








Angle of elevation

Angle of depression



Exploration Notes










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m










2400 m2650 m

d

Strait ofGibraltar

Gibraltar

Figure 5-5d


cos adjacent

__________ hypotenuse 2400 ____ 2650





d 2650 sin 250876deg 11236102



SOLUTION

473 m

h

53deg

Figure 5-5c


EXAMPLE 2


SOLUTION







Notes








Watertower

53

473 m Figure 5-5b


EXAMPLE 1


271









5182099182

CAS Suggestions














Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e



PRO B LE M N OTES


Q3 z __ x Q4 x __ z

Q5 y __ x Q6

y __ z

Q7 __3____2 Q8 2

__2____2








Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m







Sun

Angle ofelevation

Flagpole36deg116 m

Figure 5-5h





10 miNorthRim South


13deg

Figure 5-5i



Problem Set 5-5

Reading Analysis


Quick Review


called a





y

90deg90deg

1

1

Figure 5-5f

y

180deg0deg

1

1

Figure 5-5g







5min

xz

yFigure 5-5e


273


9b 744



13 254



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l







14b 1914c

15a 34515b 101cm


Altitude

Distance


710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m



710 cmLevel

Pavement

144 cm

Figure 5-5m





Distance


Figure 5-5n








deg

deg

Figure 5-5o









83 cm

57 cmOrgan

Tumor

Source

Gamma-ray beam

Skin

Figure 5-5j


Pine St

Market St

Front St

Figure 5-5k






CiboloCreek

276deg

How far

Surroundingland

378 m

Figure 5-5l





275






5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p






24c 583

24d sec5150 16182

__51_______2 16180

24e Answerswillvary








5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




5 - 6









Ground

2

HandleReel

Hosey

07

Figure 5-6a

y

1

1

360deg180deg

2

27

13

3

Figure 5-6b


Review Problems



55 in

Hinge

Where

90degLid

Prop 28 in














__ 5 1 ______ 2


51deg50ʹ

230 m

Figure 5-5p


277






TE ACH I N G


Section Notes



PRO B LE M N OTES







y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test





























v

uref

v

u

ref

R2c uref576v

u

ref



y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




y

28

7

5

3

164Time (h)

Dep

th (

)

Figure 5-6c





360deg

3

y

Figure 5-6d






Chapter Test










Chapter Test




















2

y

180deg R4a 210306R4b sin15051__2cos15052

__3____2

tan15052 1____

__3cot15052

__3

sec15052 2____

__3csc15052

R4c 2 __2 R4d 23_____

___

34

R4e 22 R4f 232__3










R3a sinu5 7_____

___74cosu52 5_____

___

74

R3b sin160503420cos1605209396uref520

v

u

ref


1

1

y

360deg

y cos

y sin



y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m




y

90deg90deg

1

1

Figure 5-6e


x


y4 mi21deg


3 m

28 mB


67 cm

z18deg


24 cm6 cm

A


107 mGround

193 m

Buried treasure

33deg

Figure 5-6f





y

2

2360deg 720deg

Figure 5-6g






T15 61155T16 2168165cm

T17 755224T18 161863mT19 105115m


Trigonometric and Periodic Functions · Chapter 5 begins Unit 2: Trigonometric and Periodic...

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Transcript of Trigonometric and Periodic Functions · Chapter 5 begins Unit 2: Trigonometric and Periodic...