Math30-1 Workbook Two

Book TwoTrigonometry OneTrigonometry TwoPermutations and Combinations

Mathematics 30-1

Copyright © 2014 | www.math30.ca

A workbook and animated series by Barry Mabillard

This page has been left blank for correct workbook printing.

Trigonometry IFormula Sheet

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The Unit Circle

Note: The unit circle is NOT included on the official formula sheet.

Trigonometry II

Exponential andLogarithmic Functions

Permutations &Combinations

Transformations& Operations

Polynomial, Radical& Rational Functions

Curriculum AlignmentMath 30-1: Alberta | Northwest Territories | NunavutPre-Calculus 12: British Columbia | YukonPre-Calculus 30: SaskatchewanPre-Calculus 40S: Manitoba

Mathematics 30-1Formula Sheet

Unit 1: Polynomial, Radical, and Rational FunctionsLesson 1: Polynomial FunctionsLesson 2: Polynomial DivisionLesson 3: Polynomial FactoringLesson 4: Radical Functions

Unit 2: Transformations and Operations

Lesson 5: Rational Functions ILesson 6: Rational Functions II

Lesson 1: Basic TransformationsLesson 2: Combined TransformationsLesson 3: InversesLesson 4: Function Operations

Unit 3: Exponential and Logarithmic FunctionsLesson 5: Function Composition

Lesson 1: Exponential FunctionsLesson 2: Laws of LogarithmsLesson 3: Logarithmic Functions

Unit 4: Trigonometry ILesson 1: Degrees and RadiansLesson 2: The Unit CircleLesson 3: Trigonometric Functions ILesson 4: Trigonometric Functions II

Unit 5: Trigonometry IILesson 5: Trigonometric EquationsLesson 6: Trigonometric Identities ILesson 7: Trigonometric Identities II

Unit 6: Permutations and CombinationsLesson 1: PermutationsLesson 2: CombinationsLesson 3: The Binomial Theorem

7:45 (16 days)1:38 (3 days)1:29 (3 days)1:13 (3 days)0:52 (2 days)

4:38 (11 days)

1:00 (2 days)1:33 (3 days)

0:57 (2 days)0:50 (2 days)0:42 (2 days)0:48 (2 days)

5:55 (12 days)1:21 (3 days)

1:52 (4 days)2:11 (4 days)1:52 (4 days)

9:59 (17 days)2:22 (4 days)2:15 (4 days)2:24 (5 days)1:58 (4 days)



40:19 (78 days)Total Course

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Table of Contents

Mathematics 30-1

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TrigonometryLESSON ONE - Degrees and Radians

Lesson Notes

Example 1 Define each term or phrase and draw a sample angle.

a) angle in standard position:

c) reference angle:

210°×π

180°= 7π

6

b) positive and negative angles:

Draw a standardposition angle, θ.

Draw θ = 120°

Find the referenceangle of θ = 150°.

Draw θ = -120°

Angle Definitions

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TrigonometryLESSON ONE - Degrees and RadiansLesson Notes

210°×π

180°=

7π6

d) co-terminal angles:

e) principal angle:

f) general form of co-terminal angles:

Draw the first positive co-terminal angle of 60°.

Find the principalangle of θ = 420°.

Find the first fourpositive co-terminalangles of θ = 45°.

Find the first fournegative co-terminalangles of θ = 45°.

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Lesson Notes210°×

π180°

=7π6

Three Angle Types: Degrees, Radians, and Revolutions.

a) Define degrees, radians, and revolutions.

Degrees:

Radians:

Revolutions:

Draw θ = 1°

Draw θ = 1 rad

Draw θ = 1 rev

Angle Typesand ConversionMultipliers

i)

ii)

iii)

Example 2

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210°×π

180°=

7π6

c) Contrast the decimal approximation of a radian with the exact value of a radian.

b) Use conversion multipliers to answer the questions and fill in the reference chart.

degree radian revolution

degree

radian

revolution

23° × = ________ rad

23° × = _______ rev

2.6 × = _______ °

2.6 × = _______ rev

0.75 rev × = _______ °

0.75 rev × = _______ rad

45° × = ________ rad

45° × = ________ rad

i) Decimal Approximation:

ii) Exact Value:

Conversion Multiplier Reference ChartRound all decimals to the nearest hundredth.

i)

ii)

iii)

iv)

v)

vi)

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Lesson Notes210°×

π180°

=7π6

Convert each angle to the requested form.

a) convert 175° to an approximate radian decimal.

b) convert 210° to an exact-value radian.

d) convert 2.5 to degrees.

e) convert to degrees.3π2

f) write as an approximate radian decimal.3π2

g) convert to an exact-value revolution.π2

h) convert 0.5 rev to degrees.

i) convert 3 rev to radians.

c) convert 120° to an exact-value revolution.

Round all decimals to the nearest hundredth.Angle ConversionPractice

Example 3

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210°×π

180°=

7π6

The diagram shows commonly used degrees. Find exact-value radians that correspond to each degree. When complete, memorize the diagram.

30° =

45° =

60° =

0° =

330° =

315° =300° =

90° =

= 270°

= 120°

= 135°

= 150°

= 180°

= 210°

= 225°= 240°

360° =

a) Method One: Find all exact-value radians using a conversion multiplier.

b) Method Two: Use a shortcut. (Counting Radians)

Commonly UsedDegrees and Radians

Example 4

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Draw each of the following angles in standard position. State the reference angle.

a) 210°

b) -260°

c) 5.3

d)5π4


Lesson Notes210°×

π180°

=7π6

-

e)12π

7

Reference AnglesExample 5

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210°×π

180°=

7π6

Draw each of the following angles in standard position. State the principal and reference angles.

a) 930°

b) -855°

c) 9

d)10π3

-

Principal andReference Angles

Example 6

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Lesson Notes210°×

π180°

=7π6

For each angle, find all co-terminal angles within the stated domain.

a) 60°, Domain: -360° ≤ θ < 1080° b) -495°, Domain: -1080° ≤ θ < 720°

c) 11.78, Domain: -2π ≤ θ < 4π d)8π3

, Domain: 13π2

37π5

≤ θ <

Co-terminal AnglesExample 7

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210°×π

180°=

7π6

For each angle, use estimation to find the principal angle.

a) 1893° b) -437.24

c) d)912π15

95π6

Principal Angleof a Large Angle

Example 8

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Lesson Notes210°×

π180°

=7π6

Use the general form of co-terminal angles to find the specified angle.

a) principal angle = 300° (find co-terminal angle 3 rotations counter-clockwise)

c) How many rotations are required to find the principal angle of -4300°? State the principal angle.

b)2π5

(find co-terminal angle 14 rotations clockwise)

principal angle =

d) How many rotations are required to find

the principal angle of ?32π3

State the principal angle.

General Form ofCo-terminal Angles

Example 9

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210°×π

180°=

7π6

sinθ =yr

cosθ =xr

tanθ =yx

cscθ =

secθ =

cotθ =x

yr

In addition to the three primary trigonometric ratios (sinθ, cosθ, and tanθ), there are threereciprocal ratios (cscθ, secθ, and cotθ). Given a triangle with side lengths of x and y, anda hypotenuse of length r, the six trigonometric ratios are as follows:

θ

a) If the point P(-5, 12) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.

ry

rx

xy

1cosθ

1sinθ

1tanθ

=

=

=

b) If the point P(2, -3) exists on the terminal arm of an angle θ in standard position, determine the exact values of all six trigonometric ratios. State the reference angle and the standard position angle.

Example 10 Six Trigonometric Ratios

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Lesson Notes210°×

π180°

=7π6

Determine the sign of each trigonometric ratio in each quadrant.

a) sinθ b) cosθ c) tanθ

d) cscθ e) secθ f) cotθ

g) How do the quadrant signs of the reciprocal trigonometric ratios (cscθ, secθ, and cotθ) compare to the quadrant signs of the primary trigonometric ratios (sinθ, cosθ, and tanθ)?

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

3

55

5 5

3

-3

-3

-4-4

44

θ

θ

θ

θ

Example 11 Signs of Trigonometric Ratios

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210°×π

180°=

7π6

Given the following conditions, find the quadrant(s) where the angle θ could potentially exist.

i) sinθ < 0 ii) cosθ > 0 iii) tanθ > 0

i) sinθ > 0 and cosθ > 0 ii) secθ > 0 and tanθ < 0 iii) cscθ < 0 and cotθ > 0

i) sinθ < 0 and cscθ ii) and cscθ < 0 iii) secθ > 0 and tanθ = 1=12

a)

b)

c)

Example 12 What Quadrant(s) is the Angle in?

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Lesson Notes210°×

π180°

=7π6

b)

Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a radian.

a)

Example 13 Exact Values ofTrigonometric Ratios

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210°×π

180°=

7π6

b)

Given one trigonometric ratio, find the exact values of the other five trigonometric ratios. State the reference angle and the standard position angle, to the nearest hundredth of a degree.

a)

Example 14 Exact Values ofTrigonometric Ratios

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Lesson Notes210°×

π180°

=7π6

Calculating θ with a calculator.

b) Given the point P(-4, 3), Mark tries to find the reference angle using a sine ratio, Jordan tries tofind it using a cosine ratio, and Dylan tries to find it using a tangent ratio. Why does each personget a different result from their calculator?

θ

P(-4, 3)

Mark’s Calculation of θ (using sine)

Jordan’s Calculation of θ (using cosine)

Dylan’s Calculation of θ (using tan)

sinθ =35

θ = 36.87°

cosθ =-45

θ = 143.13°

tanθ =3-4

θ = -36.87°

a) When you solve a trigonometric equation in your calculator, the answer you get for θ can seem unexpected. Complete the following chart to learn how the calculator processes yourattempt to solve for θ.

If the angle θ could exist in either quadrant ___ or ___ ...

The calculator alwayspicks quadrant

I or III or IIII or IVII or IIIII or IVIII or IV

Example 15 Calculator Concerns

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210°×π

180°=

7π6

a) Derive the formula for arc length, a = rθ. ar

θ

a

5 cm

153°

b) Solve for a, to the nearest hundredth.

The formula for arc length is a = rθ, where a is the arc length, θ is the central angle in radians, and r is the radius of the circle.The radius and arc length must have the same units.

c) Solve for θ.

θ3 cm

6 cm

d) Solve for r, to the nearest hundredth.

1.23π cm

r π2

e) Solve for n.

5π cm

6 cmn

(express your answer as a degree, to the nearest hundredth.)

(express your answer as an exact-value radian.)

Example 16 Arc Length

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Lesson Notes210°×

π180°

=7π6

Area of a circle sector.

c)

r

4 cm240°

b)

e)d)

9 cm

6 cm 3 cm

2π3

3 cm

120°

60°

7π6

θ

r2θ2

A = a) Derive the formula for the area of a circle sector, .

In parts (b - e), find the area of each shaded region.

Sector AreaExample 17

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210°×π

180°=

7π6

The formula for angular speed is , where ω (Greek: Omega)

is the angular speed, ∆θ is the change in angle, and ∆T is thechange in time. Calculate the requested quantity in each scenario.Round all decimals to the nearest hundredth.

∆θ

a) A bicycle wheel makes 100 complete revolutions in 1 minute. Calculate the angular speed in degrees per second.

b) A Ferris wheel rotates 1020° in 4.5 minutes. Calculate the angular speed in radians per second.

Example 18

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Lesson Notes210°×

π180°

=7π6

c) The moon orbits Earth once every 27 days. Calculate the angular speed in revolutions per second.If the average distance from the Earth to the moon is 384 400 km, how far does the moon travel in one second?

d) A cooling fan rotates with an angular speed of 4200 rpm. What is the speed in rps?

e) A bike is ridden at a speed of 20 km/h, and each wheel has a diameter of 68 cm. Calculate the angular speed of one of the bicycle wheels and express the answer using revolutions per second.

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210°×π

180°=

7π6

A satellite orbiting Earth 340 km above the surface makes onecomplete revolution every 90 minutes. The radius of Earth isapproximately 6370 km.

a) Calculate the angular speed of the satellite. Express youranswer as an exact value, in radians/second.

b) How many kilometres does the satellite travel in one minute? Round your answer to the nearest hundredth of a kilometre.

340 km

6370 km

Example 19

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TrigonometryLESSON TWO - The Unit Circle

Lesson Notes

Example 1

a) A circle centered at the origin can be represented by the relation x2 + y2 = r2, where r is the radius of the circle. Draw each circle:

i) x2 + y2 = 4 ii) x2 + y2 = 49

Introduction to Circle Equations.

(cosθ, sinθ)

-10

-10

10

10-10

-10

10

10

b) A circle centered at the origin with a radius of 1 has the equation x2 + y2 = 1. This special circleis called the unit circle. Draw the unit circle and determine if each point exists on the circumference of the unit circle.

i) (0.6, 0.8)

ii) (0.5, 0.5)

1

1

-1

-1

Equation ofa Circle

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TrigonometryLESSON TWO - The Unit CircleLesson Notes

(cosθ, sinθ)

c) Using the equation of the unit circle, x2 + y2 = 1, find the unknown coordinate of each point.Is there more than one unique answer?

iii) (-1, y)

i) ii) , quadrant II.

, cosθ > 0.iv)

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Lesson Notes(cosθ, sinθ)

The Unit Circle.

The following diagram is called the unit circle. Commonly used angles are shown as radians,and their exact-value coordinates are in brackets. Take a few moments to memorize this diagram.When you are done, use the blank unit circle on the next page to practice drawing the unitcircle from memory.

questions on next page.

The Unit CircleExample 2

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(cosθ, sinθ)

b) Draw the unit circle from memory using a partially completed template.

a) What are some useful tips to memorize the unit circle?

0

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Use the unit circle to find the exact value of each trigonometric ratio.

a) sin b) cos 180° c) cos 3π4

d) sin 11π6

e) sin 0 π2f) cos g) sin 4π

3 h) cos -120°

Use the unit circle to find the exact value of each trigonometric ratio.

a) cos 420° b) -cos 3π c) sin 13π6

d) cos 2π3

e) sin 9π4f) -sin g) cos2 (-840°) h) cos5π

27π3

π6

Finding Primary TrigonometricRatios with the Unit Circle

Example 3

Example 4

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(cosθ, sinθ)

Other Trigonometric Ratios.

The unit circle contains values for cosθ and sinθ only. The other four trigonometric ratios can be obtained using the identities on the right.

Given angles from the first quadrant of the unit circle, find the exact values of secθ and cscθ.

a) secθ

sec =2π

sec =3π

sec =4π

sec =6π

sec 0 =

tanθ =sinθcosθ

cotθ =cosθsinθ

secθ =1

cosθcscθ =

1sinθ

=1

tanθ

b) cscθ

csc =2π

csc =3π

csc =4π

csc =6π

csc 0 =

Other Trigonometric RatiosExample 5

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a) tanθ

tan =2π

tan =3π

tan =4π

tan =6π

tan 0 =

b) cotθ

cot =2π

cot =3π

cot =4π

cot =6π

cot 0 =

Other Trigonometric Ratios.

tanθ =sinθcosθ

cotθ =cosθsinθ

secθ =1

cosθcscθ =

1sinθ

=1

tanθ

The unit circle contains values for cosθ and sinθ only. The other four trigonometric ratios can be obtained using the identities on the right.

Given angles from the first quadrant of the unit circle, find the exact values of tanθ and cotθ.

Example 6 Other Trigonometric Ratios

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(cosθ, sinθ)

Use symmetry to fill in quadrants II, III, and IV for each unit circle.

sec 0 = 1

sec =6π

sec =4π

sec =3π

sec = undefined2π

a) secθ

csc 0 = undefined

csc =6π

csc =4π

csc =3π

csc = 02π

b) cscθ

tan 0 = 0

tan =6π

tan =4π

tan =3π

tan = undefined2π

c) tanθ

cot 0 =undefined

cot =6π

cot =4π

cot =3π

cot = 02π

d) cotθ

1 1

Symmetry of the Unit CircleExample 7

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Find the exact value of each trigonometric ratio.

a) sec 120° b) sec c) csc π3

d) csc 3π4

e) tan 5π4f) -tan g) cot2(270°) h) cotπ

65π6

3π2

Finding Reciprocal TrigonometricRatios with the Unit Circle

Example 8

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(cosθ, sinθ)

Find the exact value of each trigonometric expression.

a) b)

c) d)

Evaluating Complex Expressionswith the Unit Circle

Example 9

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Find the exact value of each trigonometric expression.

a) b)

c) d)

Evaluating Complex Expressionswith the Unit Circle

Example 10

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(cosθ, sinθ)

Find the exact value of each trigonometric ratio.

a) b)

c) d)

Finding the Trigonometric Ratiosof Large Angles with the Unit Circle

Example 11

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a) b)

c) d)

e) f)

g) h)

2

Verify each trigonometric statement with a calculator. Note: Every question in this example has already been seen earlier in the lesson.

Evaluating Trigonometric Ratios with a Calculator

Example 12

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(cosθ, sinθ)

a) What is meant when you are asked to find on the unit circle?

c) How does a half-rotation around the unit circle change the coordinates?

d) How does a quarter-rotation around the unit circle change the coordinates?2π3

b) Find one positive and one negative angle such that P(θ) =

e) What are the coordinates of P(3)? Express coordinates to four decimal places.

If θ = , find the coordinates of the point halfway around the unit circle.π6

If θ = , find the coordinates of the point a quarter-revolution (clockwise) around the unit circle.

Answer each of the following questions related to the unit circle.

Coordinate Relationshipson the Unit Circle

Example 13

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b) How is the central angle of the unit circle related to its corresponding arc length?

a) What is the circumference of the unit circle?

P(θ) = c) If a point on the terminal arm rotates from P(θ) = (1, 0) to ,what is the arc length?

d) What is the arc length from point A to point B on the unit circle?

θ

θ

θ

A

B


Circumference and ArcLength of the Unit Circle

Example 14

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(cosθ, sinθ)

b) Which trigonometric ratios are restricted to a range of -1 ≤ y ≤ 1? Which trigonometricratios exist outside that range?

a) Is sinθ = 2 possible? Explain, using the unit circle as a reference.

cosθ & sinθ

cscθ & secθ

tanθ & cotθ

Range Number Line


Domain and Rangeof the Unit Circle

Example 15

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d) If exists on the unit circle, how can the equation of the unit circle be used to find sinθ?

How many values for sinθ are possible?

e) If cosθ = 0, and 0 ≤ θ < π, how many values for sinθ are possible?

c) If exists on the unit circle, how can the unit circle be used to find cosθ?

How many values for cosθ are possible?

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(cosθ, sinθ)

a) Use the Pythagorean Theorem to prove that the equation of the unit circle is x2 + y2 = 1.

b) Prove that the point where the terminal arm intersects the unit circle, P(θ), has coordinates of (cosθ, sinθ).

c) If the point θ exists on the terminal arm of a unit circle, find the exact values

of the six trigonometric ratios. State the reference angle and standard position angle to the nearest hundredth of a degree.

Complete the following questions related to the unit circle.

Unit Circle ProofsExample 16

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In a video game, the graphic of a butterfly needs to be rotated. To make the butterfly graphic rotate, the programmer uses the equations:

to transform each pixel of the graphic from its original coordinates, (x, y), to its new coordinates, (x’, y’). Pixels may have positive or negative coordinates.

b) If a particular pixel has the coordinates (640, 480) after a rotation of , what were the original coordinates? Round coordinates to the nearest whole pixel.

a) If a particular pixel with coordinates of (250, 100) is rotated by , what are the new coordinates? Round coordinates to the nearest whole pixel.

x’ = x cos θ - y sin θ

y’ = x sin θ + y cos θ

5π4

π6

Example 17

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(cosθ, sinθ)

θA

x

h

θB

A B

From the observation deck of the Calgary Tower, an observer has to tilt their head θA down to see point A, and θB down to see point B.

b) If θA = , θB = , and x = 212.92 m, how high is the observation deck above the ground,

a) Show that the height of the observationx

cotθA - cotθB

deck is h = .

to the nearest metre?

Example 18

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TrigonometryLESSON THREE - Trigonometric Functions I

Lesson Notes

Example 1

0 π 2ππ2π θ

y

0 π 2ππ2π

20

-20

θ

y

5

-5

a)

b)

y = asinb(θ - c) + d

Label all tick marks in the following grids and state the coordinates of each point.

TrigonometricCoordinate Grids

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Lesson Notesy = asinb(θ - c) + d

0 8π8π θ

y

0 4π4π

40

-40

θ

y

c)

d)

-12

12

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Exploring the graph of y = sinθ.

π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

Unit Circle Reference

b) State the amplitude.

c) State the period.

d) State the horizontal displacement (phase shift).

e) State the vertical displacement.

f) State the θ-intercepts. Write your answer usinga general form expression.

g) State the y-intercept.

h) State the domain and range.

a) Draw y = sinθ.

Example 2 y = sinθ

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π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y


Exploring the graph of y = cosθ.

b) State the amplitude.







a) Draw y = cosθ.

Example 3 y = cosθ

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π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

33

3

33

5π3


tan 0 = 0

tan =6π

tan =4π

tan =3π

tan = undefined2π

1

tan =6

11π

tan =47π

1

tan =35π

tan = undefined23π

-

-

-

tan =32π -

tan =43π

1-

tan =6

5π -

tan =4

5π1

tan =3

4π

tan =6

7π

tan π = 0

b) Is it correct to say a tangent graph has an amplitude?







Exploring the graph of y = tanθ.

a) Draw y = tanθ.

Example 4 y = tanθ

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0

-5

5

0

-5

5

0

-5

5

0

-5

5

Graph each function over the domain 0 ≤ θ ≤ 2π.The base graph is provided as a convenience.

a) y = 3sinθ b) y = -2cosθ

The a Parameter

c) y = sinθ12

d) y = cosθ52

Example 5

2π

2π 2π

2π

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0

-28

28

0

-5

5

0

-1

1

c) write a cosine function.

Determine the trigonometric function corresponding to each graph.

a) write a sine function. b) write a sine function.

d) write a cosine function.

0

-8

8

14

π,( )

Example 6 The a Parameter

2π

2π2π

2π

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0

-5

5

0

-5

5

0

-5

5

d) y = cosθ -12

12

2π

2π 2π

2π

a) y = sinθ - 2

0

-5

5

b) y = cosθ + 4

c) y = sinθ + 212

Graph each function over the domain 0 ≤ θ ≤ 2π.The base graph is provided as a convenience.

-

The d ParameterExample 7

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0

-35

35

0

-32

32

2π

a) write a sine function.


b) write a cosine function.


0

-4

4

d) write a sine function.

0

-4

4

The d ParameterExample 8

2π

2π 2π

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0

-1

1

0

-1

1

0

-1

1

0

-1

1

10π8π6π4π2π6π2π 4π

2ππ2ππ

a) y = cos2θ b) y = sin3θ

c) y = cos θ d) y = sin θ13

15

Graph each function over the stated domain.The base graph is provided as a convenience.

(0 ≤ θ ≤ 2π) (0 ≤ θ ≤ 2π)

(0 ≤ θ ≤ 6π) (0 ≤ θ ≤ 10π)

Example 9 The b Parameter

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0

-1

1

c) y = 2cos θ - 1 (-2π ≤ θ ≤ 2π)

6π2π 4π2π-2π π-π

b) y = 4cos2θ + 6


a) y = -sin(3θ) (-2π ≤ θ ≤ 2π) (-2π ≤ θ ≤ 2π)

d) y = sin θ43

(0 ≤ θ ≤ 6π)12

2π-2π π-π 2π-2π π-π

-12

12

6

-6

-3

3

1

-1

-2

2

-1

1

Example 10 The b Parameter

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0

-2

2

a) write a cosine function.

0

-4

4

b) write a cosine function.


12π6π

2ππ

The b ParameterExample 11

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0

-2

2

0

-1

1


c) write a sine function.

2ππ

9π6π3π

The b Parameter

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d) (-2π ≤ θ ≤ 2π)

a) b)

c)


(-4π ≤ θ ≤ 4π) (-4π ≤ θ ≤ 4π)

(-2π ≤ θ ≤ 2π)

2π-2π π-π2π-2π π-π

4π-4π 2π-2π4π-4π 2π-2π

-1

1

-1

1

-4

4

-1

1

The c ParameterExample 12

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d) (-2π ≤ θ ≤ 2π)

a) b)

c)


2πθπ2

(-2π ≤ θ ≤ 6π)

(-π ≤ θ ≤ 4π)

2π-2π π-π4π2π-π

-2

2

1

-1

6π-2π 4π2π2πππ2

-4

4

-1

1

-1

1

The c ParameterExample 13

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0

-6

6

b) write a sine function.


4π2π

2πππ2

-1

1

Example 14 The c Parameter

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d) write a cosine function.

-4

4

8π-8π 4π-4π

The c Parameterc) write a sine function.

-1

1

2ππ-π

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0

-5

5

0

-1

1

0

-5

5

0

-6

6

6π2π 4π

2ππ2ππ

2ππ

a) b)

c) d)


(0 ≤ θ ≤ 6π) (0 ≤ θ ≤ 2π)

(0 ≤ θ ≤ 2π) (0 ≤ θ ≤ 2π)3-

Example 15 a, b, c, & d

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-2

2

4π2ππ 3π-2π -π

2π-2π

-12

12

π-π

a)

b)

Write a trigonometric function for each graph.Example 16 a, b, c, & d

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Exploring the graph of y = secθ.

Unit Circle Reference (for secθ)

b) State the period.

d) Write the general equation of the asymptotes.

c) State the domain and range.

a) Draw y = secθ.

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

e) Given the graph of f(θ) = cosθ, draw y = .f(θ)1

Graphing Reciprocal Functions

0 2π2π θ

y

-3

3

sec 0 = 1

sec =6π

sec =4π

sec = 23π

sec = undefined2π

sec =6

11π

sec =47π

sec = 235π

sec = undefined23π

sec =32π -2

sec =43π -

sec =6

5π -

sec = -4

5π

sec = -23

4π

sec = -6

7π

sec π = -1

Example 17

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Exploring the graph of y = cscθ.

a) Draw y = cscθ.




e) Given the graph of f(θ) = sinθ, draw y = .f(θ)1

Unit Circle Reference (for cscθ)

0 2π2π θ

y

-3

3

csc 0 = undefined

csc = 26π

csc =4π

csc =3π

csc = 12π

csc = -26

11π

csc = -47π

csc = -35π

csc = -123π

csc =32π

csc =43π

csc = 26

5π

csc = -4

5π

csc = -3

4π

csc = -26

7π

csc π =undefined

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

Example 18 Graphing Reciprocal Functions

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Exploring the graph of y = cotθ.

Unit Circle Reference (for cotθ)

a) Draw y = cotθ.

π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

3

33

33




e) Given the graph of f(θ) = tanθ, draw y = .f(θ)1

0 2π2π θ

y

-3

3

cot 0 =undefined

cot =6π

cot =4π

cot =3π

cot = 02π

11

cot =6

11π

cot =47π

1

cot =35π

cot = 023π

-

-

-

cot =32π -

cot =43π -

cot =6

5π -

cot =4

5π1

cot =3

4π

cot =6

7π

cot π =undefined

Example 19 Graphing Reciprocal Functions

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0

-3

3

2ππ

0

-3

3

0

-3

3

2ππ

y = secθ

0

-3

3

y = cscθ

2ππ 2ππ

y = cotθ

a) b)

c) d)

Graph each function over the domain 0 ≤ θ ≤ 2π. The base graph is provided as a convenience. State the new domain and range.

y = secθ

Transformations of Reciprocal Functions

Example 20

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TrigonometryLESSON FOUR - Trigonometric Functions II

Lesson Notes

h(t)

t

Example 17Example 1 Trigonometric Functions of Angles

a) b)

-2

2

y = cosθ(one cycle shown)

π 2π 3π

ii) Graph this function using technology. ii) Graph this function using technology.

θ

y

1

-1

-2

2

y = cosθ(one cycle shown)

180º θ

y

1

-1

i) Graph: i) Graph:

360º 540º

(0 ≤ θ < 3π) (0º ≤ θ < 540º)

TrigonometricFunctions of Angles

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Lesson Notes

h(t)

t

b)

-2

2

x

y

1

-1

88 16 24

Trigonometric Functions of Real Numbers.

ii) Graph this function using technology.

i) Graph:

a)

-2

2

t

h

1

-1

15 30 45 60

ii) Graph this function using technology.

i) Graph:

c) What are three differences between trigonometric functions of angles and trigonometric functions of real numbers?

Example 17Example 2 Trigonometric Functions of Real Numbers

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Lesson Notes

h(t)

t

a)

Determine the view window for each function and sketch each graph.

b)

Example 17Example 3 Graph Preperationand View Windows

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Lesson Notes

h(t)

t

a)

b)

Example 17Example 4 Determine the view window for each function and sketch each graph.

Graph Preperationand View Windows

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Lesson Notes

h(t)

t

0

-10

10

168


-5

5

168-4

b) write a sine function.

Determine the trigonometric functioncorresponding to each graph.Example 17Example 5 Find the Trigonometric

Function of a Graph

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Lesson Notes

h(t)

t

0

10

25

-10

(8, 9)

(16, -3)


0

300

2400

-300

(300, -50)

(1425, 150)


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Lesson Notes

h(t)

t

c) If the range of y = 3cosθ + d is [-4, k], determine the values of d and k.

a) If the transformation g(θ) - 3 = f(2θ) is applied to the graph of f(θ) = sinθ, find the new range.

Answer the following questions:

b) Find the range of4

.

Example 17Example 6 Assorted Questions

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Lesson Notes

h(t)

t

d) State the range of f(θ) - 2 = msin(2θ) + n.

e) The graphs of f(θ) and g(θ) intersect at the points If the amplitude of each graph is quadrupled, determine the new points of intersection.

and

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Lesson Notes

h(t)

t

b) Find the y-intercept of

Answer the following questions:

a) If the point lies on the graph of , find the value of a.

.

Example 17Example 7 Assorted Questions

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Lesson Notes

h(t)

t

d) The graph of f(θ) = kcosθ is transformed to the graph of g(θ) = bcosθ by a vertical stretch about the x-axis.

If the point

state the vertical stretch factor.

π2

π 3π2

2π

f(θ)

g(θ)

k

b

c) The graphs of f(θ) and g(θ) intersect at the point (m, n). Find the value of f(m) + g(m).

f(θ)

g(θ)(m, n)

m

n

exists on the graph of g(θ),

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Lesson Notes

h(t)

t

12 cm

4 cm

8 cm

2 s1 s 3 s 4 s0 cm

ground level

The graph shows the height of a pendulum bob as a function of time. One cycle of a pendulum consists of two swings - a right swing and a left swing.

t

h(t)

a) Write a function that describes the height of the pendulum bob as a function of time.

b) If the period of the pendulum is halved, how will this change the parameters in the function you wrote in part (a)?

c) If the pendulum is lowered so its lowest point is 2 cm above the ground, how will this changethe parameters in the function you wrote in part (a)?

Example 17Example 8

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Lesson Notes

h(t)

t

t

h(t)

A wind turbine has blades that are 30 m long. An observer notes that one blade makes 12 complete rotations (clockwise) every minute. The highest point of the blade during the rotation is 105 m.

A

a) Using Point A as the starting point of the graph, draw the height of the blade over two rotations.

b) Write a function that corresponds to the graph.

c) Do we get a different graph if the wind turbine rotates counterclockwise?

Example 17Example 9

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Lesson Notes

h(t)

t

θ

h

150 m

A person is watching a helicopter ascend from a distance 150 m away from the takeoff point.

a) Write a function, h(θ), that expresses the height as a function of the angle of elevation. Assume the height of the person is negligible.

b) Draw the graph, using an appropriate domain.

c) Explain how the shape of the graph relates to the motion of the helicopter.

θ

h(θ)

Example 17Example 10

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Lesson Notes

h(t)

t

c) Calculate the height of the mass after 1.2 seconds. Round your answer to the nearest hundredth.

a) Draw the graph for two full oscillations of the mass.

d) In one oscillation, how many seconds is the mass lower than 3.2 m? Round your answer to the nearest hundredth.

A mass is attached to a spring 4 m above the ground and allowed to oscillate from its equilibrium position. The lowest position of the mass is 2.8 m above the ground, and it takes 1 s for one complete oscillation.

t

h(t)

b) Write a sine function that gives the height of the mass above the ground as a function of time.


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Lesson Notes

h(t)

t

c) Calculate the height of the rider after 1.6 rotations of the Ferris wheel.Round your answer to the nearest hundredth.

a) Draw the graph for two full rotations of the Ferris wheel.

d) In one rotation, how many seconds is the rider higher than 26 m? Round your answer to the nearest hundredth.

A Ferris wheel with a radius of 15 m rotates once every 100 seconds. Riders board the Ferris wheel using a platform 1 m above the ground.

t

h(t)

b) Write a cosine function that gives the height of the rider as a function of time.


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Lesson Notes

h(t)

t

b) Draw the graph for one complete cycle (winter solstice to winter solstice).

December 216h, 46m

March 2112h, 17m

June 2117h, 49m

September 2112h, 17m

December 216h, 46m

a) Convert each date and time to a number that can be used for graphing.

December 21 = March 21 = June 21 = September 21 = December 21 =

The following table shows the number of daylight hours in Grande Prairie.

n

d(n)

Day Number

Daylight Hours 6h, 46m = 12h, 17m = 17h, 49m = 12h, 17m = 12h, 46m =


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Lesson Notes

h(t)

t

c) Write a cosine function that relates the number of daylight hours, d, to the day number, n.

e) In one year, approximately how many days have more than 17 daylight hours?Round your answer to the nearest day.

d) How many daylight hours are there on May 2? Round your answer to the nearest hundredth.

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Lesson Notes

h(t)

t

The highest tides in the world occur between New Brunswick and Nova Scotia, in the Bay of Fundy. Each day, there are two low tides and two high tides. The chart below contains tidal height data that was collected over a 24-hour period.

a) Convert each time to a decimal hour.

Time Decimal Hour Height of Water (m)

2:12 AM

8:12 AM

2:12 PM

8:12 PM

3.48

13.32

3.48

13.32

Low Tide

Low Tide

High Tide

High Tide

Bay ofFundy

b) Graph the height of the tide for one full cycle (low tide to low tide).

t

h(t)


Note: Actual tides at the Bay of Fundy are 6 hours and 13 minutes apart dueto daily changes in the position of the moon.

In this example, we willuse 6 hours for simplicity.

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Lesson Notes

h(t)

t

c) Write a cosine function that relates the height of the water to the elapsed time.

d) What is the height of the water at 6:09 AM? Round your answer to the nearest hundredth.

e) For what percentage of the day is the height of the water greater than 11 m?Round your answer to the nearest tenth.

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Lesson Notes

h(t)

t

a) Graph the population of owls and mice over six years.

b) Describe how the graph shows the relationship between owl and mouse populations.

A wooded region has an ecosystem that supports both owls and mice. Owl and mice populations vary over time according to the equations:

Owl population:

Mouse population:

where O is the population of owls, M is the population of mice, and t is the time in years.

Population

Time(years)

200

250

300

8000

12000

16000

1 2 3 4 5 60

Mic

eO

wls

7 8


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Lesson Notes

h(t)

t

The angle of elevation between the 6:00 positionand the 12:00 position of a historical building’s clock, as measured from an observer

The observer also knows that he is standing 424 m away from the clock, and his eyes are at the same height as the base of the clock.The radius of the clock is the same as the length of the minute hand.

If the height of the minute hand’s tip is measured relative to the bottom of the clock, what is the height of the tip at 5:08,to the nearest tenth of a metre?

π444

π444

.standing on a hill, is

424 m


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Lesson Notes

h(t)

t

Shane is on a Ferris wheel, and his height can be described

Tim, a baseball player, can throw a baseball with a speed of 20 m/s. If Tim throws a ball directly upwards, the height can be determined by the equation hball(t) = -4.905t2 + 20t + 1

If Tim throws the baseball 15 seconds after the ride begins, when are Shane and the ball at the same height?

by the equation .


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TrigonometryLESSON FIVE - Trigonometric Equations

Lesson Notes

Example 1

a)

Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation. Write the general solution.

b)

c) 0 d) tan2θ = 1

Solving equations with the unit circle.

Primary Ratios

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TrigonometryLESSON FIVE - Trigonometric EquationsLesson Notes


a)

-1

1

2ππ

sinθ b)

-1

1

2ππ

sinθ = -1

c)

-1

1

2ππ

cosθd) cosθ = 2

-2

2

1

-1

2ππ

e) tanθ

-2

2

2ππ

1

-1

3

-3

f) tanθ = undefined

-2

2

2ππ

1

-1

3

-3

Primary Ratios

Solving equations graphically withintersection points

Example 2

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Lesson Notes

Find all angles in the domain 0° ≤ θ ≤ 360° that satisfy the given equation. Write the general solution.

a)

b)

c)

360°

0°

90°

180°

270°

360°

0°

90°

180°

270°

360°

0°

90°

180°

270°

Solving equations with a calculator. (degree mode)

Primary RatiosExample 3

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Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation.

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

b) cosθ =

a) sinθ = 1

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Intersection Point(s)of Original Equation θ-Intercepts

θ-InterceptsIntersection Point(s)of Original Equation

Solving equations graphically withθ-intercepts.

Primary RatiosExample 4

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Lesson Notes

Solve

a) non-graphically, using the cos-1 feature of a calculator.

b) non-graphically, using the unit circle.

c) graphically, using the point(s) of intersection.

-3

3

-2

-1

2

1

2ππ

d) graphically, using θ-intercepts.

-

-3

3

-2

-1

2

1

2ππ

0 ≤ θ ≤ 2π Primary Ratios

Equations withprimary trig ratios

Example 5

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a) non-graphically, using the sin-1 feature of a calculator.




-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Solve sinθ = -0.30 θ ε R Primary Ratios

Equations withprimary trig ratios

Example 6

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Lesson Notes


Solving equations with the unit circle.

a)

c)

b)

Reciprocal RatiosExample 7

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Find all angles in the domain 0 ≤ θ ≤ 2π that satisfy the given equation.Write the general solution.

b)

d)

f)

-2

2

1

-1

2ππ

Reciprocal Ratios

Solving equations graphically withintersection points

θ

θa) θ

-2

2

1

-1

2ππ

c) θ

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

e) θ

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

secθ = -1

Example 8

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Lesson Notes

b)

a)

Find all angles in the domain 0°≤ θ ≤ 360°that satisfy the given equation. Write the general solution

360°

0°

90°

180°

270°

c)

360°

0°

90°

180°

270°

360°

0°

90°

180°

270°

Solving equations with a calculator. (degree mode)

Reciprocal RatiosExample 9

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Solving equations graphically withθ-intercepts.

Reciprocal Ratios

a) θ

b) θ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Intersection Point(s)of Original Equation θ-Intercepts

θ-InterceptsIntersection Point(s)of Original Equation

Example 10

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Lesson Notes

a) non-graphically, using the sin-1 feature of a calculator.



-3

3

-2

-1

2

1

2ππ


-3

3

-2

-1

2

1

2ππ

Solve cscθ = -2 0 ≤ θ ≤ 2π Reciprocal Ratios

Equations withreciprocal trig ratios

Example 11

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a) non-graphically, using the cos-1 feature of a calculator.




-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Solve secθ = -2.3662 Reciprocal Ratios

Equations withreciprocal trig ratios

0°≤ θ ≤ 360°Example 12

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Lesson Notes


b)

d) 4secθ + 3 = 3secθ + 1c) 3tanθ - 5 = 0

a) cosθ - 1 = 0 θ

First-Degree TrigonometricEquations

Example 13

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a) 2sinθcosθ = cosθ b) 7sinθ = 4sinθ

d) tanθ + cosθtanθ = 0c) sinθtanθ = sinθ

Check the solution graphically.

-1

1

2ππ


-1

1

2ππ


-1

1

2ππ


-1

1

2ππ

First-Degree TrigonometricEquations

Example 14

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Lesson Notes


a) sin2θ = 1 b) 4cos2θ - 3 = 0

c) 2cos2θ = cosθ d) tan4θ - tan2θ = 0


-1

1

2ππ


-1

1

2ππ


-3

3

-2

-1

2

1

2ππ


-1

1

2ππ

Second-DegreeTrigonometricEquations

Example 15

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a) 2sin2θ - sinθ - 1 = 0

b) csc2θ - 3cscθ + 2 = 0

c) 2sin3θ - 5sin2θ + 2sinθ = 0


-3

3

-2

-1

2

1

2ππ


-3

3

-2

-1

2

1

2ππ


-3

3

-2

-1

2

1

2ππ

Second-DegreeTrigonometricEquations

Example 16

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Lesson Notes

Solve each trigonometric equation.

a)

ii) non-graphically:i) graphically:

b)


θ

θ

0 ≤ θ ≤ 2π

0 ≤ θ ≤ 2π

-1

1

2ππ

-1

1

π2

π 2π3π2

Example 17 Double and Triple Angles

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Solve each trigonometric equation.

a)


b)


θ

θ

0 ≤ θ ≤ 4π

0 ≤ θ ≤ 8π

-1

1

4π2ππ 3π

-1

1

8π4π2π 6π

-1

Example 18 Half and Quarter Angles

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Lesson Notes

It takes the moon approximately 28 days to go through all of its phases.

a) Write a function, P(t), that expresses the visible percentage of the moon as a function of time.Draw the graph.

NewMoon

FirstQuarter

FullMoon

LastQuarter

NewMoon

b) In one cycle, for how many days is 60% or more of the moon’s surface visible?

1.00

28147 210

0.50

t

Visible %

Example 19

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a) Write a tangent function, d(θ), that expresses the distance where the water splashes the wall as a function of therotation angle θ.

b) Graph the function for one complete rotation of the sprinkler. Draw only the portion of the graph that actually corresponds to the wall being splashed.

A rotating sprinkler is positioned 4 m away from the wall of a house. The wall is 8 m long. As the sprinkler rotates, the stream of water splashes the house d meters from point P.Note: North of point P is a positive distance, and south of point P is a negative distance.

c) If the water splashes the wall 2.0 m north of point P, what is the angle of rotation (in degrees)?

-8

8

2πππ2

3π2

4

-4

θ

d

Pθ

d

N

S

W E

Rotating SprinklerExample 20

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Lesson Notes

In this example, we will explore the inverse functions of sine and cosine to learn why taking an inverse actually yields the solution.

When we solve a trigonometric equation like cosx = -1, one possible way to write the solution is:

Inverse Trigonometric Functions

x

y

0 π2

π2

π

π

2π

2π

3π2

3π2

π2 π

2

π3π2

2π

π

3π2

2π

1

1

-1

-2

-3

-4

-5

-6

2

2

3

3

4

4

5

5

6

6-6 -5 -4 -3 -2 -1

a) When we draw the inverse of trigonometric graphs, it is helpful to use a grid that is labeled with both radians and integers. Briefly explain how this is helpful.

Example 21Enrichment Example

Students who plan on taking university calculus should complete this example.

Inverse Trigonometric Functions

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b) Draw the inverse function of each graph. State the domain and range of the original and inverse graphs (after restricting the domain of the original so the inverse is a function).

y = sinx y = cosx

x

y

0 π2

π2

π

π

2π

2π

3π2

3π2

π2 π

2

π3π2

2π

π

3π2

2π

1

1

-1

-2

-3

-4

-5

-6

2

2

3

3

4

4

5

5

6

6-6 -5 -4 -3 -2 -1

x

y

0 π2

π2

π

π

2π

2π

3π2

3π2

π2 π

2

π3π2

2π

π

3π2

2π

1

1

-1

-2

-3

-4

-5

-6

2

2

3

3

4

4

5

5

6

6-6 -5 -4 -3 -2 -1

c) Is there more than one way to restrict the domain of the original graph so the inverse is a function? If there is, generalize the rule in a sentence.

d) Using the inverse graphs from part (b), evaluate each of the following:

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TrigonometryLESSON SIX - Trigonometric Identities I

Lesson Notes

Example 1 Understanding Trigonometric Identities.

a) Why are trigonometric identities considered to be a special type of trigonometric equation?

b) Which of the following trigonometric equations are also trigonometric identities?

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

A trigonometric equation that ISan identity:

A trigonometric equation that is NOT an identity:

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

i) ii) iii)

iv) v)

TrigonometricIdentities

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TrigonometryLESSON SIX- Trigonometric Identities ILesson Notes

The Pythagorean Identities.

a) Using the definition of the unit circle, derive the identity sin2x + cos2x = 1. Why is sin2x + cos2x = 1 called a Pythagorean Identity?

b) Verify that sin2x + cos2x = 1 is an identity using i) x = and ii) x = .

c) Verify that sin2x + cos2x = 1 is an identity using a graphing calculator to draw the graph.

-1

1

2ππ

sin2x + cos2x = 1

Example 2 PythagoreanIdentities

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Lesson Notes

d) Using the identity sin2x + cos2x = 1, derive 1 + cot2x = csc2x and tan2x + 1 = sec2x.

e) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities for x = .

f) Verify that 1 + cot2x = csc2x and tan2x + 1 = sec2x are identities graphically.

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

1 + cot2x = csc2x tan2x + 1 = sec2x

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Prove that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

a)

b)

Reciprocal IdentitiesExample 3

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

NOTE: You will need to use a graphing calculator to obtain the graphs in this lesson. Make sure the calculator is in RADIAN mode, and use window settings that match the grid provided in each example.

Rewrite the identity so it is absolutely true.(i.e. Include restrictions on the variable)


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Lesson Notes

a)

b)


Example 4 Reciprocal Identities

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1



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a)

b)

Pythagorean IdentitiesProve that each trigonometric statement is an identity. State the non-permissible values of x so the identity is true.

-1

1

2ππ

-1

1

2ππ



Example 5

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Lesson Notes

c)

d)

-1

1

2ππ

-1

1

2ππ



Pythagorean Identities

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a)

b)


-1

1

2ππ

-1

1

2ππ



Example 6 Pythagorean Identities

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Lesson Notes

c)

d)

0

2

2ππ

1

-1

1

2ππ



Pythagorean Identities

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Common Denominator

Proofsa)

b)


-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1



Example 7

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Lesson Notes

c)

d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1



Common Denominator

Proofs

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a)

b)


-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1



Example 8 Common Denominator

Proofs

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Lesson Notes

c)

d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1



Common Denominator

Proofs

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AssortedProofs

a) b)

c) d)

Prove each identity. For simplicity, ignore NPV’s and graphs.Example 9

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Lesson Notes

a) b)

c) d)

Prove each identity. For simplicity, ignore NPV’s and graphs.Example 10 Assorted

Proofs

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a) b)

c) d)

Prove each identity. For simplicity, ignore NPV’s and graphs.Example 11 Assorted

Proofs

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Lesson Notes

c) State the non-permissible values for

Exploring the proof of

a) Prove algebraically that b) Verify that for

d) Show graphically thatAre the graphs exactly the same?

π3

y1 = sinx

y2 = tanxcosx

Example 12

-1

1

2ππ

-1

1

2ππ

. .

.

Exploringa Proof

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c) State the non-permissible values



d) Show graphically that

π3

for Are the graphs exactly the same?

y1 =

y2 =

Example 13

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

. .

.

Exploringa Proof

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Lesson Notes

c) State the the non-permissible values



d) Show graphically that

π2

forAre the graphs exactly the same?

Example 14

.

.

.

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

y1 =

y2 =

Exploringa Proof

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Solve each trigonometric equationover the domain 0 ≤ x ≤ 2π.

a) b)

c) d)

Equations With Identities

-3

3

2ππ

-2

-1

2

1

-6

6

2ππ

-4

-2

4

2

-3

3

2ππ

-2

-1

2

1

-1

1

2ππ

Example 15

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Lesson Notes

a) b)

c) d)

Equations With IdentitiesSolve each trigonometric equationover the domain 0 ≤ x ≤ 2π.

-3

3

2ππ

-2

-1

2

1

-2

2

2ππ

-10

10

2ππ

-1

1

2ππ

Example 16

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a) b)

c) d)

Equations With IdentitiesSolve each trigonometric equationover the domain 0 ≤ x ≤ 2π.

-10

10

2ππ

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-12

-9

-6

3

02ππ

-3

Example 17

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Lesson Notes

Use the Pythagorean identities to find the indicated value and draw the corresponding triangle.

b) If the value of , find the value of secA within the same domain.

c) If find the exact value of sinθ. 7cosθ = , and cotθ < 0,

7

a) If the value of find the value of cosx within the same domain.

Example 18 Pythagorean Identitiesand Finding an Unknown

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Trigonometric Substitution.

3

θ

a) Using the triangle to the right, show that

b) Using the triangle to the right, show that

b

aHint: Use the triangle to find a trigonometric expression equivalent to b.

θ

4

aHint: Use the triangle to find a trigonometric expression equivalent to a.

can be expressed as .

can be expressed as .

Example 19 Trigonometric Substitution

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Example 1 Evaluate each trigonometricsum or difference.

a) b)

c) d)

TrigonometryLESSON SEVEN - Trigonometric Identities II

Lesson Notes

e) f)

Sum and Difference Identities

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TrigonometryLESSON SEVEN- Trigonometric Identities II

Lesson Notes

Write each expression as a single trigonometric ratio.

a)

c)

b)

Example 2 Sum and Difference Identities

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Lesson Notes

Find the exact value ofeach expression.

a)

b) c)

d) Given the exact values of cosine and sine for 15°, fill in the blanks for the other angles.


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a)

b)

c)


Lesson Notes

Find the exact value ofeach expression.For simplicity, do not rationalize the denominator.


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Double-angle identities.

a) Prove the double-angle sine identity, sin2x = 2sinxcosx.

b) Prove the double-angle cosine identity, cos2x = cos2x - sin2x.

c) The double-angle cosine identity, cos2x = cos2x - sin2x, can be expressed as cos2x = 1 - 2sin2x or cos2x = 2cos2x - 1. Derive each identity.

d) Derive the double-angle tan identity, .


Lesson Notes

Example 5 Double-Angle Identities

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Lesson Notes

a) Evaluate each of the following expressions using a double-angle identity.

b) Express each of the following expressions using a double-angle identity.

c) Write each of the following expression as a single trigonometric ratio using a double-angle identity.

Example 6 Double-angle identities. Double-Angle Identities

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Lesson Notes

Note: Variable restrictions may beignored for the proofs in this lesson.

a)

b)

c) d)

Prove each trigonometric identity.Example 7 Sum and Difference Identities

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Lesson Notes

a)

b)

c) d)

Example 8 Prove each trigonometric identity. Sum and Difference Identities

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a) b)

c) d)


Lesson Notes

Example 9 Prove each trigonometric identity. Double-Angle Identities

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a)

c) d)

b)


Lesson Notes

Example 10 Prove each trigonometric identity. Double-Angle Identities

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a) b)

c) d)


Lesson Notes

Example 11 Prove each trigonometric identity. Assorted Proofs

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Lesson Notes

a) b)

c) d)


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Lesson Notes

a) b)

c) d)

Solve each trigonometric equationover the domain 0 ≤ x ≤ 2π.Example 14 Assorted Equations

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Lesson Notes

a) b)

c) d)

Example 15 Solve each trigonometric equationover the domain 0 ≤ x ≤ 2π. Assorted Equations

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Lesson Notes

a) b)

c) d)


0

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Lesson Notes

a) b)

c) d)


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Lesson Notes

Trigonometric identities and geometry.A

C

B

b) If A = 32° and B = 89°, what is the value of C?

a) Show that

Example 18

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Lesson Notes

Solve for x. Round your answer to the nearest tenth.

57

176

153

104

A

B

x

Trigonometric identities and geometry.Example 19

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Lesson Notes

θIf a cannon shoots a cannonball θ degrees above the horizontal,the horizontal distance traveled by the cannonball before it hitsthe ground can be found with the function:

( )

2 sin cos4.9

ivd

θ θθ =

a) Rewrite the function so it involves a single trigonometric identity.

b) Graph the function. Use the graph to describe the trajectory of the cannonball at the following angles: 0°, 45°, and 90°.

c) If the cannonball travels a horizontal distance of 100 m, find the angle of the cannon. Solve graphically, and round your answer to the nearest tenth of a degree.

The initial velocity of the cannonball is 36 m/s.

d

θ

Example 20

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Lesson Notes

An engineer is planning the construction of a road through a tunnel. In one possible design, the width of the road maximizes the area of a rectangle inscribed within the cross-section of the tunnel.

The angle of elevation from the centre line of the road to the upper corner of the rectangle is θ.Sidewalks on either side of the road are included in the design.

a) If the area of the rectangle can be represented by the function A(θ) = msin2θ, what is the value of m?

θ

70 m

road widthsidewalk sidewalk

A

θ

b) What angle maximizes the area of the rectangular cross-section?

c) For the angle that maximizes the area:i) What is the width of the road? ii) What is the height of the tallest vehicle that will pass through the tunnel? iii) What is the width of one of the sidewalks?Express answers as exact values.

Example 21

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Lesson Notes

a) Two sound waves are represented with f(θ) and g(θ).

i) Draw the graph of y = f(θ) + g(θ) and determine the resultant wave function.ii) Is this constructive or destructive interference?iii) Will the new sound be louder or quieter than theoriginal sound?

0

-6

6

2π

f(θ) = 2cosθ

π

g(θ) = 4cosθ

The improper placement of speakers for a home theater system may result in a diminished sound quality at the primary viewing area.This phenomenon occurs because sound waves interact with each other in a process called interference. When two sound waves undergo interference, they combine to form a resultant sound wave that has an amplitude equal to the sum of the component sound wave amplitudes.

If the amplitude of the resultant wave is larger than the component wave amplitudes, we say the component waves experienced constructive interference.

If the amplitude of the resultant wave is smaller than the component wave amplitudes, we say the component waves experienced destructive interference.

Example 22

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Lesson Notes

b) A different set of sound waves are represented with m(θ) and n(θ).

i) Draw the graph of y = m(θ) + n(θ) and determine the resultant wave function. ii) Is this constructive or destructive interference?iii) Will the new sound be louder or quieter than theoriginal sound?

0

-6

6

2π

m(θ) = 2cosθ

π

n(θ) = 2cos(θ - π)

c) Two sound waves experience total destructive interference if the sum of their wave functions is zero. Given p(θ) = sin(3θ - 3π/4) and q(θ) = sin(3θ - 7π/4), show that these waves experience total destructive interference.

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Lesson Notes

Even & Odd Identities

a) Explain what is meant by the terms even function and odd function.

Even & Odd Identities

b) Explain how the even & odd identities work. (Reference the unit circle or trigonometric graphs in your answer.)

c) Prove the three even & odd identities algebraically.

Example 23

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Lesson Notes

βα

A BE G

F

C

D

1

Proving the sum and difference identities.

a) Explain how to construct the diagram shown.

b) Explain the next steps in the construction.

βα

A BE G

FH

C

D

1

α

Example 24 Enrichment ExampleStudents who plan on taking university calculus should complete this example.

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Lesson Notes

d) Prove the sum and difference identity for sine.

c) State the side lengths of all the triangles.

FH

D

α

E

1

A

D

βα+

1

β

A

F

D

αA G

F

α

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Permutations and CombinationsLESSON ONE - Permutations

Lesson NotesnPr =

n!(n - r)!

Example 1

a) Use a tree diagram to find the number of unique words.

b) Use the Fundamental Counting Principle to find the number of unique words.

c) Use permutation notation to find the number of unique words. Evaluate using a calculator.

Three letters (A, B, and C) are taken from a set of letter tiles and arranged to form “words”. In this question, ACB counts as a word - even though it’s not an actual English word. A B

C

Permutations

d) What is meant by the terms single-case permutation and multi-case permutation?

e) Use permutations to find the number of ways a one-, two-, or three-letter word can be formed.

Introduction to Permutations.

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Permutations and CombinationsLESSON ONE - PermutationsLesson Notes

nPr =n!

(n - r)!

Evaluate each of the following factorial expressions.

a) b) 1!

5!3!

4! c) 0!

e) f)n!

(n - 2)!g)

8!7!•2!

d) (-2)!

h)(n + 1)!(n - 1)!

FactorialNotation

Example 2

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Lesson NotesnPr =

n!(n - r)!

a) A Grade 12 student is taking Biology, English, Math, and Physics in her first term. If a student timetable has room for five courses (meaning the student has a spare), how many ways can she schedule her courses?

b) A singing competition has three rounds. In each round, the singer has to perform one song from a particular genre. How many different ways can the performer select the genres?

c) A web development team of three members is to be formed from a selection pool of 10 people. The team members will be assigned roles of programmer, graphic designer, and database analyst. How many unique teams are possible? You can assume that each person in the selection pool is capable of performing each task.

Round 1 Round 2 Round 3

RockMetalPunk

Alternative

PopDance

CountryBluesFolk

Block 1Block 2Block 3Block 4Block 5

CourseMath 30-1

SparePhysics 30

English 30-1Biology 30

BlockOne Possible Timetable

d) There are 13 letter tiles in a bag, and no letter is repeated. Using all of the letters from the bag, a six-letter word, a five-letter word, and a two-letter word are made. How many ways can this be done?

i) Fundamental Counting Principle ii) Permutation Notation




Single-CasePermutations

Simple PermutationsPermutations with Repetitions NOT Allowed.(Finite Sample Sets)

Example 3

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nPr =n!

(n - r)!

a) How many ways can the letters in the word SEE be arranged?

i) Tree Diagram ii) Fundamental Counting Principle

b) How many ways can the letters in the word MISSISSAUGA be arranged?

c) A multiple-choice test has 10 questions. Three questions have an answer of A, four questions have an answer of B, one question has an answer of C, and two questions have an answer of D. How many unique answer keys are possible?

d) How many pathways exist from point A to point B if the only directions allowed are north and east?

A

B

e) How many ways can three cars (red, green, blue) be parked in five parking stalls?

f) An electrical panel has five switches. How many ways can the switches be positioned up or down if three switches must be up and two must be down?

Repetitions NOT AllowedPermutations with Repetitions NOT Allowed.(Finite Sample Sets)

Example 4 Single-CasePermutations

One possible switch arrangement.

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Lesson NotesnPr =

n!(n - r)!

a) There are three switches on an electrical panel. How many unique up/down sequences are there?

b) How many two-letter “words” can be created using the letters A, B, C, and D?

c) A coat hanger has four knobs, and each knob can be painted any color. If six different colors of paint are available, how many ways can the knobs be painted?

e) An identification code consists of any two letters followed by any three digits. How many identification codes can be created?

d) A phone number in British Columbia consists of one of four area codes (236, 250, 604, and 778),followed by a 7-digit number that cannot begin with a 0 or 1. How many unique phone numbers are there?

Repetitions ARE AllowedPermutations where Repetitions ARE Allowed.(Infinite Sample Sets)


One possible switch arrangement.

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nPr =n!

(n - r)!

Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if:

a) Frank must be seated in the third chair?

b) Brenda or Cory must be in the second chair, and Eliza must be in the third chair?

c) Danielle can’t be at either end of the line?

d) men and women alternate positions, with a woman sitting in the first chair?

e) the line starts with the pattern man-man-woman?

Constraints andLine Formations

Permutations with Repetitions NOT Allowed.(Finite Sample Sets)


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Lesson NotesnPr =

n!(n - r)!

How many ways can you order the letters from the word TREES if:

a) a vowel must be at the beginning?

b) it must start with a consonant and end with a vowel?

c) the R must be in the middle?

d) it begins with exactly one E?

e) it ends with TR?

f) consonants and vowels alternate?

Constraintsand Words

Permutations with Repetitions NOT Allowed.(Finite Sample Sets)


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nPr =n!

(n - r)!

a) How many ways can 3 chemistry books, 4 math books, and 5 physics books be arranged if books on each subject must be kept together?

b) How many arrangements of the word ACTIVE are there if C&E must always be together?

c) How many arrangements of the word ACTIVE are there if C&E must always be together, and in the order CE?

d) Six people (Andrew, Brenda, Cory, Danielle, Eliza, Frank) are going to be seated in a line. How many unique lines can be formed if Cory, Danielle, and Frank must be seated together?

Objects ALWAYS TogetherPermutations with Repetitions NOT Allowed.(Finite Sample Sets)

Single-CasePermutations

Example 8

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Lesson NotesnPr =

n!(n - r)!

a) How many ways can the letters in QUEST be arranged if the vowels must never be together?

b) Eight cars (3 red, 3 blue, and 2 yellow) are to be parked in a line. How many unique lines can be formed if the yellow cars must not be together? Assume that cars of each color are identical.

c) How many ways can the letters in READING be arranged if the vowels must never be together?

Objects NEVER TogetherPermutations with Repetitions NOT Allowed.(Finite Sample Sets)

i) Use a shortcut that works for separating two items. ii) Use a general method.

i) Use a shortcut that works for separating two items. ii) Use a general method.


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nPr =n!

(n - r)!

Example 10

a) How many words (with at most three letters) can be formed from the letter tiles SUNDAY?

c) How many 3-digit odd numbers greater than 600 can be formed using the digits 2, 3, 4, 5, 6, and 7, if a number contains no repeating digits?

More Than One Case. (At Least/At Most)

d) Six vehicles (3 different brands of cars and 3 different brands of trucks) are going to be parked in a line. How many unique lines can be formed if the row starts with at least two trucks?

Multi-CasePermutations

b) How many words (with at least five letters) can be formed from the letter tiles SUNDAY?

e) Six vehicles (3 different brands of cars and 3 different brands of trucks) are going to be parked in a line. How many unique lines can be formed if trucks and cars alternate positions?

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Lesson NotesnPr =

n!(n - r)!

a) Evaluate 4P3

b) Evaluate 12P3

c) Write as a permutation.

d) Write 3! as a permutation.

Permutation Formula: Solve each of the following without using a calculator. nPr =

n!

(n - r)!

5!

3!

Example 11

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nPr =n!

(n - r)!

Equations with Factorials and Permutations. Solve each of the following without using a calculator.

b) (n + 2)! = 12n!a)n!

(n - 2)!= 5n

d)(2n + 1)!

(2n - 1)!= 4n + 2c)

n!

10= n-1Pn-3

nPr =n!

(n - r)!Example 12

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Lesson NotesnPr =

n!(n - r)!

Equations with Factorials and Permutations. Solve each of the following without using a calculator.

a) nP2 = 56 b) 6Pr = 120

c) n + 3P2 = 20 d) n - 3P1 = 2•n - 4P1

Example 13nPr =

n!

(n - r)!

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nPr =n!

(n - r)!


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Permutations and CombinationsLESSON TWO - Combinations

Lesson Notes

Example 1There are four marbles on a table, and each marble is a different color (red, green, blue, and yellow). Two marbles are selected from the table at random and put in a bag.

a) Is the order of the marbles, or the order of their colors, important?

b) Use a tree diagram to find the number of unique color combinations for the two marbles.

c) Use combination notation to find the number of unique color combinations.

nCr =n!

(n - r)!r!

Combinations

d) What is meant by the terms single-case combination and multi-case combination?

e) How many ways can three or four marbles be chosen?

Introduction to Combinations.

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Permutations and CombinationsLESSON TWO - CombinationsLesson Notes

nCr =n!

(n - r)!r!

Example 2 Sample Sets with NO Subdivisions

a) There are five toppings available for a pizza (mushrooms, onions, pineapple, spinach, and tomatoes). If a pizza is ordered with three toppings, and no toppingmay be repeated, how many different pizzas can be created?

b) A committee of 4 people is to be formed from a selection pool of 9 people. How many possible committees can be formed?

c) How many 5-card hands can be made from a standard deck of 52 cards?

d) There are 9 dots randomly placed on a circle.i) How many lines can be formed within the circle by connecting two dots?

ii) How many triangles can be formed within the circle?

5♣ 6♥ 2♦

A 4♥

Single-CaseCombinations

Combinations with Repetitions NOT Allowed.(Finite Sample Sets)

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Lesson NotesnCr =

n!(n - r)!r!

Example 3

a) How many 6-person committees can be formed from 11 men and 9 women if 3 men and 3 women must be on the committee?

b) A crate of toy cars contains 10 working cars and 4 defective cars. How many ways can 5 cars be selected if only 3 work?

c) From a deck of 52 cards, a 6-card hand is dealt. How many distinct hands are there if the hand must contain 2 spades and 3 diamonds?

d) A bouquet contains four types of flowers:

A florist is making a bouquet that uses one type of focal flower, no fragrant flowers, three types of line flowers and all of the filler flowers. How many different bouquets can be made?

Focal Flowers: Large and eye-catching flowers that draw attention to one area of the bouquet.

Roses, Peonies, Hydrangeas, Chrysanthemums, Tulips, and Lilies

Fragrant Flowers: Flowers that add a pleasant fragrance to the bouquet.

Petunia, Daffodils, Daphnes, Gardenia, Lilacs, Violets, Magnolias

Line Flowers: Tall and narrow flowers used to establish the height of a floral bouquet.

Delphiniums, Snapdragons, Bells of Ireland, Gladioli, and Liatris

Filler Flowers: Unobtrusive flowers that give depth to the bouquet.

Daisies, Baby's Breath, Wax Flowers, Solidago, and Caspia

Flower Type Examples

Sample Sets with SubdivisionsCombinations with Repetitions NOT Allowed.(Finite Sample Sets)


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Example 4


nCr =n!

(n - r)!r!

a) A committee of 5 people is to be formed from a selection pool of 12 people. If Carmen must be on the committee, how many unique committees can be formed?

b) A committee of 6 people is to be formed from a selection pool of 11 people. If Grant and Helen must be on the committee, but Aaron must not be on the committee, how many unique committees can be formed?

d) From a deck of 52 cards, a 5-card hand is dealt. How many distinct 5-card hands are there if the ace of spades and two of diamonds must be in the hand?

e) A lottery ticket has 6 numbers from 1-49. Duplicate numbers are not allowed, and the order of the numbers does not matter. How many different lottery tickets contain the numbers 12, 24 and 48, but exclude the numbers 30 and 40?

c) Nine students are split into three equal-sized groups to work on a collaborative assignment.How many ways can this be done? Does the sample set need to be subdivided in this question?

More Sample Sets with Subdivisions



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Example 5


Lesson NotesnCr =

n!(n - r)!r!

a) How many five-letter words using letters from TRIANGLE can be made if the five-letter word must have two vowels and three consonants?

b) There are 4 men and 5 women on a committee selection pool. A three-person committee consisting of President, Vice-President, and Treasurer is being formed. How many ways can exactly two men be on the committee?

c) A music teacher is organizing a concert for her students. If there are six piano students andseven violin students, how many different concert programs are possible if four piano studentsand three violin students perform in an alternating arrangement?

Permutations and Combinations Together



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nCr =n!

(n - r)!r!

Example 6

a) Twelve people at a party shake hands once with everyone else in the room. How many handshakes took place?

c) If there are 8 dots on a circle, how many quadrilaterals can be formed?

d) A polygon has 6 sides. How many diagonals can be formed?

Handshakes, Teams,and Shapes.


b) If each of the 8 teams in a league must play each other three times, how many games will be played? (Note: This is a multi-case combination)


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Lesson NotesnCr =

n!(n - r)!r!

Example 7 Combinations where Repetitions ARE Allowed.(Infinite Sample Sets)

a) A jar contains quarters, loonies, and toonies. If four coins are selected from the jar, how many unique coin combinations are there?

b) A bag contains marbles with four different colors (red, green, blue, and yellow).If three marbles are selected from the bag, how many unique color combinations are there?


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nCr =n!

(n - r)!r!

Example 8 More Than One Case (At Least/At Most).

a) A committee of 5 people is to be formed from a group of 4 men and 5 women. How many committees can be formed if at least 3 women are on the committee?

b) From a deck of 52 cards, a 5-card hand is dealt. How many distinct hands can be formed if there are at most 2 queens?

Multi-CaseCombinations

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Lesson NotesnCr =

n!(n - r)!r!

d) A research team of 5 people is to be formed from 3 biologists, 5 chemists, 4 engineers, and 2 programmers. How many teams have exactly one chemist and at least 2 engineers?

e) In how many ways can you choose one or more of 5 different candies?

c) From a deck of 52 cards, a 5-card hand is dealt. How many distinct hands can be formed if there is at least 1 red card?

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nCr =n!

(n - r)!r!

Example 9 Combination Formula.

a) Evaluate 7C5

b) Evaluate 3C3

nCr =n!

(n - r)!r!

c) Evaluate

d) Write as a combination.6!

4!2!

e) Write as a combination.5!

4!

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Lesson NotesnCr =

n!(n - r)!r!

Example 10 Combination Formula.Solve for the unknown algebraically.

a) nC2 = 21 b) 4Cr = 6

c) d)

nCr =n!

(n - r)!r!

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nCr =n!

(n - r)!r!

Example 11 Combination Formula.Solve for the unknown algebraically.

a) b)

c) d)

nCr

nCn - r

= 1nC4

n - 2C2

= 1

n - 1P3 = 2 × n - 1C2 12

× n + 2C3n + 1C2 =

nCr =n!

(n - r)!r!

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Lesson NotesnCr =

n!(n - r)!r!

Example 12 Assorted Mix I

a) A six-character code has the pattern shown below, and the same letter or digit may be used more than once. How many unique codes can be created?

Letter LetterDigit Digit Digit Digit

b) If there are 2 different parkas, 5 different scarves, and 4 different tuques, how many winter outfits can be made if an outfit consists of one type of each garment?

c) If a 5-card hand is dealt from a deck of 52 cards, how many hands have at most one diamond?

d) If there are three cars and four motorcycles, how many ways can the vehicles park in a linesuch that cars and motorcycles alternate positions?

Assorted Mix I

STRATEGY: Organize your thoughts with these guiding questions:

1) Permutation or Combination?2) Single-Case or Multi-Case?3) Are repetitions allowed?4) What is the sample set?Are there subdivisions?5) Are there any tricks or shortcuts?

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nCr =n!

(n - r)!r!

e) Show that nCr =nCn - r.

f) There are nine people participating in a raffle. Three $50 gift cards from the same store are to be given out as prizes. How many ways can the gift cards be awarded?

g) There are nine competitors in an Olympic event. How many ways can the bronze, silver, and gold medals be awarded?

h) A stir-fry dish comes with a base of rice and the choice of five toppings: broccoli, carrots, eggplant, mushrooms, and tofu. How many different stir-fry dishes can be prepared if thecustomer can choose zero or more toppings?

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Lesson NotesnCr =

n!(n - r)!r!

Example 13

a) A set of tiles contains eight letters, A - H. If two of these sets are combined, how manyways can all the tiles be arranged? Leave your answer as an exact value.

b) A pattern has five dots such that no three points are collinear. How many lines can be drawn if each dot is connected to every other dot?

c) How many ways can the letters in CALGARY be arranged if L and G must be separated?

d) A five-person committee is to be formed from 11 people. If Ron and Sara must be included, but Tracy must be excluded due to a conflict of interest, how many committees can be formed?

Assorted Mix II Assorted Mix II

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nCr =n!

(n - r)!r!

e) Moving only south and east, how many unique pathwaysconnect points A and C?

f) How many ways can the letters in SASKATOON be arranged if the letters K and T mustbe kept together, and in that order?

A

C

B

g) A 5-card hand is dealt from a deck of 52 cards. How many hands are possible containingat least three hearts?

h) A healthy snack contains an assortment of four vegetables. How many ways can one or more of the vegetables be selected for eating?

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Lesson NotesnCr =

n!(n - r)!r!

Example 14

a) How many ways can the letters in EDMONTON be arranged if repetitions are not allowed?

b) A bookshelf has n fiction books and six non-fiction books. If there are 150 ways to choose twobooks of each type, how many fiction books are on the bookshelf?

c) How many different pathways exist between points A and D?

d) How many numbers less than 60 can be made using only the digits 1, 5, and 8, if the numbers formed may contain repeated digits?

Assorted Mix III

A

B

D

C

Assorted Mix III

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nCr =n!

(n - r)!r!

e) A particular college in Alberta has a list of approved pre-requisite courses:

f) How many ways can four bottles of different spices be arranged on a spice rack with holesfor six spice bottles?

g) If there are 8 rock songs and 9 pop songs available, how many unique playlists containing3 rock songs and 2 pop songs are possible?

h) A hockey team roster contains 12 forwards, 6 defencemen, and 2 goalies. During play,only six players are allowed on the ice - 3 forwards, 2 defencemen, and 1 goalie. How manydifferent ways can the active players be selected?

Five courses are required for admission to the college. Math 30-1 (or Math 30-2) and English 30-1 are mandatory requirements, and at least one science course must be selected as well. How many different ways could a student select five courses on their college application form?

Math Science English Other

Math 30-1or

Math 30-2

Biology 30Chemistry 30Physics 30

English 30-1 Option AOption BOption COption DOption E

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Lesson NotesnCr =

n!(n - r)!r!

c) If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all thesame color?

Example 15a) A fruit mix contains blueberries, grapes, mango slices, pineapple slices, and strawberries. If six pieces of fruit are selected from the fruit mix and put on a plate, how many ways can this be done?

b) How many ways can six letter blocks be arranged in a pyramid, if all of the blocks are used?

A

B C

FED

d) If a 5-card hand is dealt from a deck of 52 cards, how many hands have cards that are all thesame suit?

Assorted Mix IV Assorted Mix IV

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nCr =n!

(n - r)!r!

e) A multiple choice test contains 5 questions, and each question has four possible responses.How many different answer keys are possible?

f) How many diagonals are there in a pentagon?

g) How many ways can eight books, each covering a different subject, be arranged on a shelf such that books on biology, history, or programming are never together?

h) If a 5-card hand is dealt from a deck of 52 cards, how many hands have two pairs?

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Lesson NotesnCr =

n!(n - r)!r!

Example 16

d) In genetics, a codon is a sequence of three letters that specifies a particular amino acid. A fragment of a particular protein yields the aminoacid sequence:

Met - Gly - Ser - Arg - Cys - Gly.

How many unique codon arrangements could yield this amino acid sequence?

Arginine (Arg)

Cysteine (Cys)

Glycine (Gly)

Methionine (Met)

Serine (Ser)

CGU, CGC, CGA, CGG, AGA, AGG

UGU, UGC

GGU, GGC, GGA, GGG

AUG

UCU, UCC, UCA, UCG, AGU, AGC

Amino Acid Codon(s)

Assorted Mix V

c) Five different types of fruit and six different types of vegetables are available for a healthy snack tray. The snack tray is to contain two fruits and three vegetables. How many different snack trays can be made if blueberries or carrots must be served, but not both together?

b) Show that 25! + 26! = 27 × 25!

a) How many ways can six people be split into two equal-sized groups?

Assorted Mix V

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nCr =n!

(n - r)!r!

f) The discount shelf in a bookstore has a variety of books on computers, history, music, and travel. The bookstore is running a promotion where any five books from the discount shelf can be purchased for $20. How many ways can five books be purchased?

g) Show that nCr + nCr + 1 = n + 1Cr + 1.

h) How many pathways are there from point A to point C, passing through point B? Each step of the pathway must be getting closer to point C.

e) In a tournament, each player plays every other player twice. If there are 56 games, how many people are in the tournament?

A

B

C

Note: This question will require more paper than is provided on this page.

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Permutations and CombinationsLESSON THREE - The Binomial Theorem

Lesson Notestk+1 = nCk(x)n-k(y)k

Example 1Pascal’s Triangle is a number pattern withuseful applications in mathematics.Each row is formed by adding together adjacent numbers from the preceding row. First seven rows

of Pascal’s Triangle.

1

1 1

1 12

11 3 3

1 14 6 4

1 15 10 10 5

1 16 15 61520

a) Determine the eighth row of Pascal’s Triangle.

b) Rewrite the first seven rows of Pascal’s Triangle, but use combination notation instead of numbers.

c) Using the triangles from parts (a & b) as a reference, explain what is meant by nCk = nCn - k.

Pascal’s Triangle Pascal’s Triangle

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Permutations and CombinationsLESSON THREE - The Binomial TheoremLesson Notes

tk+1 = nCk(x)n-k(y)k

Rows and Terms of Pascal’s Triangle.

a) Given the following rows from Pascal’s Triangle, write the circled number as a combination.

b) Use a combination to find the third term in row 22 of Pascal’s Triangle.

c) Which positions in the 12th row of Pascal’s Triangle have a value of 165?

d) Find the sum of the numbers in each of the first four rows of Pascal’s Triangle. Use your result to derive a function, S(n), for the sum of all numbers in the nth row of Pascal’s Triangle. What is the sum of all numbers in the eleventh row?

1 8 28 56 70 56 28 8 1i)

1 12 66 220 495 792 924 792 495 220 66 12 1ii)

Pascal’s TriangleExample 2

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Use Pascal’s Triangle to determine the number of paths from point A to point B if east and south are the only possible directions.

a) b)

c) d)

A

B

A

B

A

B

A

B

Pascal’s Triangleand Pathways

Example 3

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b) (x + 2)6

c) (2x - 3)4

The Binomial Theorem.

a) Define the binomial theorem and explain how it is used to expand (x + 1)3.

Expand the expressions in parts (b) and (c) using the binomial theorem.



The Binomial TheoremExample 4

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Expand each expression.

a) (x2 - 2y)4

b)

c)



The Binomial TheoremExample 5

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Write each expression as a binomial power.

a) x4 + 4x3y + 6x2y2 + 4xy3 + y4

b) 32a5 - 240a4b + 720a3b2 - 1080a2b3 + 810ab4 - 243b5

c)

Condense to a Binomial

Example 6

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a) Find the third term in the expansion of (x - 3)4 .

Use the general term formula to find the requested term in a binomial expansion.

b) Find the fifth term in the expansion of (3a3 - 2b2)8 .

c) Find the fourth term in the expansion of . 6

2 1x -

x

tk + 1 = nCk(x)n - k(y)k

General TermExample 7

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Answer each of the following questions.

a) In the expansion of (5a - 2b)9, what is the coefficient of the term containing a5 ?

b) In the expansion of (4a3 + 3b3)5, what is the coefficient of the term containing b12 ?

c) In the expansion of (3a - 4)8, what is the middle term?

d) If there are 23 terms are in the expansion of (a - 2)3k-5, what is the value of k?

Finding Specific ValuesExample 8

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c) A term in the expansion of (a + m)7 is . What is the value of m.

a) A term in the expansion of (ma - 4)5 is -5760a2. What is the value of m?

b) The term -1080a2b3 occurs in the expansion of (2a - 3b)n. What is the value of n?

ba

Finding Specific ValuesExample 9 Answer each of the following questions.

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b) In the expansion of , what is the constant term?

a) In the expansion of , what is the constant term?

c) In the expansion of , one of the terms is 240x2. What is the value of b?b

Finding Specific ValuesExample 10 Answer each of the following questions.

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Answer KeyTrigonometry Lesson One: Degrees and Radians Note: For illustrative purposes,

all diagram angles will be in degrees.

120°

-120°

θ

a) The rotation angle between the initial arm and the terminal arm is called the standard position angle.

Example 1:b) An angle is positive if we rotate the terminal arm counter-clockwise, and negative if rotated clockwise.

c) The angle formed between the terminal arm and the x-axis is called the reference angle.

d) If the terminal arm is rotated by a multiple of 360° in either direction, it will return to its original position. These angles are called co-terminal angles.

e) A principal angle is an angle that exists between 0°and 360°.

f) The general form of co-terminal angles is θc = θp + n(360°) using degrees, or θc = θp + n(2π) using radians.

420° 60°

45°,405°, 765°, 1125°, 1485°

45°, -315°, -675°, -1035°, -1395°

57.3°

1°

Example 2:

150°

30°

a) i. One degree is defined as 1/360th of a full rotation.

ii. One radian is the angle formed when the terminal arm swipes out an arc that has the same length as the terminal arm. One radian is approximately 57.3°.

iii. One revolution is defined as 360º, or 2pi. It is one complete rotation around a circle.

b) i. 0.40 rad ii. 0.06 rev iii. 148.97° iv. 0.41 rev v. 270° vi. 4.71 rad c) i. 0.79 rad ii. π/4 rad

Conversion Multiplier Reference Chart

degree radian revolution

degree

radian

revolution

π180°

π180°

2π1 rev

2π1 rev

1 rev

360°

1 rev

360°

Example 3: a) 3.05 rad b) 7π/6 rad c) 1/3 rev d) 143.24° e) 270° f) 4.71 rad g) 1/4 rev h) 180° i) 6π rad

30° =

45° =

60° =

0° =

330° =

315° =

300° =

90° =

= 270°

= 120°

= 135°

= 150°

= 180°

= 210°

= 225°

= 240°

360° =

Example 4:

210°

30°θ

-260°

80°

θ

304°

θ

56°

135°

θ45°

309°

51°

θ

Example 5:a) θr = 30° b) θr = 80° c) θr = 56°

(or 0.98 rad)d) θr = 45°(or π/4 rad)

e) θr = 51°(or 2π/7 rad)

120°

θ

60°156°

θ24°

225°

θ45°

210°

θ30°

Example 6:a) θp = 210°, θr = 30° b) θp = 225°, θr = 45° c) θp = 156°, θr = 24° d) θp = 120°, θr = 60°

(or θp = 2.72, θr = 0.42) (or θp = 2π/3, θr = π/3)

Example 7:

a) θ = 60°, θp = 60° b) θ = -495°, θp = 225°

c) θ = 675°, θp = 315° d) θ = 480°, θp = 120°

θc = -855°, -135°, 225°, 585°θc = -300°, 420°, 780° Example 8:a) θp = 93° b) θp = 148° c) θp = 144° d) θp = 330°

(or 2.58 rad) (or 4π/5 rad) (or 11π/6 rad)

Example 9:a) θc = 1380° b) θc = -138π/5 c) θc = 20° d) θc = 2π/3

60°

225°

θc = -45°, 315° θc = -960°, -600°, -240°, 120°,840°, 1200°(or θc = -0.785, 5.50)

(or θc = -16π/3,-10π/3, -4π/3, 2π/3, 14π/3, 20π/3)

120°

315°

93°

148° 144°

330°

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Example 10:a) θp = 112.62°, θr = 67.38° b) θp = 303.69°, θr = 56.31°

Example 11:a) sinθ: QI: +, QII: +, QIII: -, QIV: -b) cosθ: QI +, QII: -, QIII: -, QIV: +c) tanθ: QI +, QII: -, QIII: +, QIV: -d) cscθ: QI: +, QII: +, QIII: -, QIV: -e) secθ: QI +, QII: -, QIII: -, QIV: +f) cotθ: QI +, QII: -, QIII: +, QIV: -g) sinθ & cscθ share the same quadrant signs.cosθ & secθ share the same quadrant signs.tanθ & cotθ share the same quadrant signs

Example 12: a) i. QIII or QIV ii. QI or QIV iii. QI or QIII b) i. QI ii. QIV iii. QIII c) i. none ii. QIII iii. QI

-12

-513

22.62°202.62°

θ

73

154.62°

25.38°

θ

Example 13:a) θp = 202.62°, θr = 22.62° b) θp = 154.62°, θr = 25.38°

(or θp = 3.54 rad, θr = 0.39 rad) (or θp = 2.70 rad, θr = 0.44 rad)

Example 14:

-2

3

33.69°

326.31°

θ

4

-35

323.13°

36.87°

θ

a) θp = 323.13°, θr = 36.87° b) θp = 326.31°, θr = 33.69°

Example 15:

If the angle θ could exist in either quadrant ___ or ___ ...

The calculator alwayspicks quadrant

I or III or IIII or IVII or IIIII or IVIII or IV

IIIIIIVIV

a)

b) Each answer is different because the calculator is unawareof which quadrant the triangle is in. The calculator assumesMark’s triangle is in QI, Jordan’s triangle is in QII, and Dylan’striangle is in QIV.

Example 16:a) The arc length can be found by multiplying the circumference by the sector percentage. This gives us:a = 2πr × θ/2π = rθ.b) 13.35 cmc) 114.59°d) 2.46 cme) n = 7π/6

Example 17:a) The area of a sector can be found by multiplying the area of the full circle by the sector percentage to get the area of the sector.This gives us:a = πr2 × θ/2π = r2θ/2.b) 28π/3 cm2

c) 3π cm2

d) 81π/2 cm2

e) 15π cm2

Example 18:a) 600°/sb) 0.07 rad/sc) 1.04 kmd) 70 rev/se) 2.60 rev/s

Example 19:a) π/2700 rad/sb) 468.45 km

θ-5

1312

112.62°

67.38°2

-356.31°

θ

303.69°

Answer Key

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Answer KeyTrigonometry Lesson Two: The Unit Circle

Example 1:

-10

10

10 -10

10

10

(0.6, 0.8)

(0.5, 0.5)

a) i. ii. b) i. Yes ii. No

Example 2: See Video.

a)Example 3: b) -1 c)

d) e) 0 f) 0 g) h)

a)Example 4: b) 1 c)

d) e) -1 f) g) h)

, , , ,

, , , ,

Example 5:

a)

b)

, , , ,

, , , ,

Example 6:

a)

b)


Example 8:

a) -2 b) undefined c) d) e) f) -1 g) 0 h)

Example 9:

a) b) 1 c) d)

Example 10:

a) 1 b) c) d)

Example 11:

a) -1 b) c) undefined d) undefined


Example 13:a) P(π/3) means "point coordinates at π/3".

b) c)

d) e) P(3) = (-0.9900, 0.1411)

Example 14:a) C = 2π b) The central angle and arc length of

the unit circle are equal to each other.c) a = 2π/3 d) a = 7π/6

Example 15:a) The unit circle and the line y = 2do not intersect, so it's impossible for sinθ to equal 2.

y = 2

b)

cosθ & sinθ

cscθ & secθ

tanθ & cotθ

Range Number Line

10-1

10-1

10-1

c) d) 53.13°, 302.70° e)

Example 16:a) Inscribe a right triangle with side lengths of |x|, |y|, and a hypotenuse of 1 into the unit circle. We use absolute values because technically, a triangle must have positive side lengths. Plug these side lengths into the Pythagorean Theorem to get x2 + y2 = 1.

|x|

|y|1

b) Use basic trigonometric ratios (SOHCAHTOA)to show that x = cosθ and y = sinθ. c) θp = 167.32°, θr = 12.68°

Example 17:a) (167, 212) b) (-792, 113)

Example 18:a) See Video b) 160 m

c) i. ii.

iii. y = 0 iv.

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Answer KeyTrigonometry Lesson Three: Trigonometric Functions I

Example 1:a) (-5π/6, 3), (-π/6, -4), (7π/6, 1) b) (-3π/4, -12), (π/4, 16), (7π/4, -8) c) (-6π, 8), (-2π, -8), (4π, -4) d) (-3π, 10), (3π/2, -30), (5π/2, -20)

Example 2: a) y = sinθ b) a = 1 c) P = 2πd) c = 0 e) d = 0 f) θ = nπ, nεI g) (0, 0)h) Domain: θ ε R, Range: -1 ≤ y ≤ 1

π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

Example 3: a) y = cosθ b) a = 1 c) P = 2πd) c = 0 e) d = 0 f) θ = π/2 + nπ, nεI g) (0, 1)h) Domain: θ ε R, Range: -1 ≤ y ≤ 1

π2

3π2

0 π 2ππ2

π3π2

2π

-1

22

32

12

12

22

32

1

π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

33

3

33

5π3

Example 4: a) y = tanθ b) Tangent graphs do not have an amplitude. c) P = π d) c = 0 e) d = 0 f) θ = nπ, nεIg) (0, 0) h) Domain: θ ε R, θ ≠ π/2 + nπ, nεI, Range: y ε R

Example 5:

a)

b)

Example 6:

c)

d)

a) b)Example 8: -c) d) 5

c) d)

a) b)

0

-5

5

2π0

-5

5

2π

0

-5

5

2π0

-5

5

2π

0

-5

5

2π0

-5

5

2π2π0

-5

5

0

-5

5

2π

Example 7:

c) d)a) b)

Example 9:

c) d)a) b)

0

-1

1

2ππ0

-1

1

2ππ0

-1

1

6π2π 4π0

-1

1

10π8π6π4π2π

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Answer KeyExample 10:

a) b)

2π-2π π-π

-1

1

2π-2π π-π

-12

12

6

-6

c) d)

2π-2π π-π

-3

3

1

-1

-2

2

0

-1

1

6π2π 4π

4π-4π 2π-2π

-1

1

4π

-4π 2π-2π

-1

1

2π-2π π-π

-1

1

2π

-2π π-π

-4

4

Example 12:

a) b)

c) d)

2πππ2

-1

1

6π-2π 4π2π

-1

1

4π2π-π

2

1

-1

Example 13:

a) b)

c) d)

2π-2π π-π

4

0

-5

5

6π2π 4π0

-1

1

2ππ

0

-5

5

2ππ0

-6

6

2ππ

Example 15:

a) b)

c) d)

Example 16:

a) b)

a) b)

Example 11:

c) d) a) b)

Example 14:

c) d)

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Answer Key

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

π 2ππ2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

-3

1

-1

3π2

π2

3π2

π2

3

3

33

33

π2

3π2

0 π 2ππ2

π3π2

2π π6

π4

π3

3π4

2π3

5π6

7π6

5π4

4π3

5π3

7π4

11π6

π6

π4

π3

2π3

3π4

5π6

7π6

5π4

4π3

5π3

7π4

11π6 θ

y

3

2

-3

22

33

-2

233

2

1

-1

Example 17: a) y = secθ b) P = 2π c) Domain: θ ε R, θ ≠ π/2 + nπ, nεI; Range: y ≤ -1, y ≥ 1d) θ = π/2 + nπ, nεI

Example 18: a) y = cscθ b) P = 2π c) Domain: θ ε R, θ ≠ nπ, nεI; Range: y ≤ -1, y ≥ 1d) θ = nπ, nεI

Example 19: a) y = cotθ b) P = π c) Domain: θ ε R, θ ≠ nπ, nεI; Range: yεRd) θ = nπ, nεI

Example 20:

Domain: θ ε R, θ ≠ π/2 + nπ, nεI;(or: θ ε R, θ ≠ π/2 ± nπ, nεW) Range: y ≤ -1/2, y ≥ 1/2

Domain: θ ε R, θ ≠ n(2π), nεI;(or: θ ε R, θ ≠ ±n(2π), nεW) Range: y ε R

Domain: θ ε R, θ ≠ π/4 + nπ/2, nεI;(or: θ ε R, θ ≠ π/4 ± nπ/2, nεW) Range: y ≤ -1, y ≥ 1

Domain: θ ε R, θ ≠ π/4 + nπ, nεI;(or: θ ε R, θ ≠ π/4 ± nπ, nεW) Range: y ≤ -1, y ≥ 1

0 2π2π θ

y

-3

3

0 2π2π θ

y

-3

3

0 2π2π θ

y

-3

3

a)

0

-3

3

2ππ

b)

0

-3

3

2ππ 0

-3

3

2ππ

d)c)

0

-3

3

2ππ

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Answer KeyTrigonometry Lesson Four: Trigonometric Functions II

Example 1:

-2

2

π 2π 3π θ

y

1

-1

-2

2

180º θ

y

1

-1

360º 540º

a) b)

-2

2

t

h

1

-1

15 30 45 60

-2

2

x

y

1

-1

88 16 24

Example 2:

a) b)

-26

4 8

y

x

-24-22-20-18-16-14-12-10-8-6-4-2

1 2 3 5 6 7

x-225-200 -150

-175 -125 -75 -25-100 -50

025

50 10075 125 175

150 200 250225 275

25

50

75y

Example 3:

a) b)

-8

y

x

-6-4-202468

10121416

120110100908070605040302010

1820

y

x

2

54321-1-2-3

468

101214161820

Example 4:

a) b)

Example 5:

a)

b)

c)

d)

Example 6:

a)

b)

c)

d)

5e)

Example 7:

a)

b)

c)

d)

Example 8:

a) b) The b-parameter is doubled when the period is halved. The a, c, and d parameters remain the same.

c) The d-parameter decreases by 2 units, giving us d = 4. All other parameters remain unchanged.

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Answer Key

t

h(t)

5 10

45

75

105

Example 9:

a)

b) c) If the wind turbine rotates counterclockwise, we still get the same graph.

Example 10:

a)

θ

h(θ)

90°

c) The angle of elevation increases quickly at first, but slows down as the helicopter reaches greater heights. The angle never actually reaches 90°.

b) ,

Example 11:

a)

b) c) 2.86 m

Example 12:

a)

c) 28.14 mb)

t

h(t)

5.2

4.0

2.8

1 2

t

h(t)

0 25 50 75 100 125 150 175 200

1

16

31

d) 0.26 s

Example 13:

b)

d) 15.86 h

n

d(n)

-50 0 50 100 150 200 250 300 350 400

4

8

12

16

20

24

( ) ( )2

5.525cos 11 12.295365

d n nπ = − + +

c)

a) Decimal daylight hours: 6.77 h, 12.28 h, 17.82 h, 12.28 h, 6.77 h

e) 64 days

Example 14:

b)

d) 10.75 m

a) Decimal hours past midnight: 2.20 h, 8.20 h, 14.20 h, 20.20 h

e) 32.3%

t

h(t)

4

8

12

16

0 4 8 12 16 20 24

c)

Population

Time(years)

200

250

300

8000

12000

16000

1 2 3 4 5 60

Mic

eO

wls

7 8

M(t)

O(t)

Example 15:

a) b) See Video.

Example 16: 2.5 m

Example 17: 15.6 s and 18.3 s

t

h(t)

1.5

15

3.0

30 45 60

(8, 2.5)

t

h(t)

10

15

19

30 45 60

(18.3, 13.1)

1

(15.6, 10.5)

d) 26.78 s

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Answer KeyTrigonometry Lesson Five: Trigonometric Equations

Example 1:

b) , c) d)

Note: n ε I for all general solutions.

Example 2:

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 3:

360°

0°

90°

180°

270°

30°30°

30°150°

360°

0°

90°

180°

270°

30°

150°

30°

210°

360°

0°

90°

180°

270°

45°

45°

45°

225°

Intersection point(s)of original equation

-3

3

-2

-1

2

1

2ππ

θ-intercepts

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Intersection point(s)of original equation

-3

3

-2

-1

2

1

2ππ

θ-intercepts

Example 4:

b)

,a)

a) b) c)

d) no solution e) f)

-2

2

2ππ

1

-1

3

-3

-2

2

2ππ

1

-1

3

-3-2

2

1

-1

2ππ

a) b) c)

a)

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60°

60°

120°

240°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 6:

a) 197.46° and 342.54° b) 197.46° and 342.54° c) 197.46° and 342.54° d) 197.46° and 342.54°

17.46°

197.46°

17.46°

342.54°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

The unit circle is not useful for this question.

Example 7:

Example 8:

a) No Solution b) c)

d) e) f)

-2

2

1

-1

2ππ

-2

2

1

-1

2ππ

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

-2

2

2ππ

1

-1

a) b) c) d)

a) b) c)

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Answer Key

Intersection point(s)of original equation θ-intercepts

Intersection point(s)of original equation θ-intercepts

Example 10:

b)a) No Solution

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 11:

a) b) c) d)

30°

210°

30°

330°

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 12:

a) b) c) d)

115°

245°

65°65°

The unit circle is not useful for this question.

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 13:

a) b) d)c)

360°

0°

90°

180°

270°

60°60°

60°120°

360°

0°

90°

180°

270°

60°

60°

240°

60°

Example 9:

a) b) c)

360°

0°

90°

180°

270°

60°

60°

120°

240°

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-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 15:

a) b) c) d)

-1

1

2ππ

-1

1

2ππ

-3

3

-2

-1

2

1

2ππ

-1

1

2ππ

Example 16:

a) b) c)

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

Example 17:

a) b) a) b)

Example 18:

Example 19:

1.00

28147 210

0.50

t

Visible %

a)

b) Approximately 12 days.

Example 21: See VideoExample 20: a)

b) See graph. c) 0.4636 rad (or 26.6°)

-8

8

2πππ2

3π2

4

-4

θ

d

0.4636

2

a) b) c) d)

-1

1

2ππ

-1

1

π2

π 2π3π2

-1

1

4π2ππ 3π

-1

1

8π4π2π 6π

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Answer KeyTrigonometry Lesson Six: Trigonometric Identities I

Example 1:


Example 2:

Example 3:

a) b)

Example 4:

a) b) -1

1

2ππ

-1

1

2ππ

-1

1

2ππ

-1

1

2ππ

Example 5:

,a) b)

c) d)

Identity

-3

3

-2

-1

2

1

2ππ

Equation

-3

3

-2

-1

2

1

2ππ

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

-2

-1

2

1

2ππ

-3

3

-2

-1

2

1

2ππ

a)

a)

b) i) ii) iii) iv) v)

Not an Identity Not an Identity Identity Identity

-1

1

2ππx

yθ

1Use basic trigonometry(SOHCAHTOA) to showthat x = cosθ and y = sinθ.

b) Verify that the L.S. = R.S. for each angle.

c) The graphs of y = sin2x + cos2x and y = 1 are the same.

d) Divide both sides ofsin2x + cos2x = 1 bysin2x to get 1 + cot2x = csc2x.Divide both sides ofsin2x + cos2x = 1 bycos2x to get tan2x + 1 = sec2x.

e) Verify that the L.S. = R.S. for each angle.

f) The graphs of y = 1 + cot2x and y = csc2x are the same.

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

The graphs of y = tan2x + 1 and y = sec2x are the same.

-3

3

2ππ

-2

-1

2

1

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

Not an Identity

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a) b)

c) d)

-1

1

2ππ

-1

1

2ππ

Example 7:

a) b)

c) d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

0

2

2ππ

1

-1

1

2ππ

Example 8:

a) b)

c) d)

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

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Answer KeyExample 9: See Video Example 10: See Video Example 11: See Video

Example 12:

a) See Video

b)

c)

d)

Example 13:

a) See Video

b)

c)

d)

Example 14:

a) See Video

b)

c)

d)

-1

1

2ππ

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

Example 15:

Example 16:

a) b)

c) d)

, ,

, ,

-3

3

2ππ

-2

-1

2

1

-3

3

2ππ

-2

-1

2

1

-6

6

2ππ

-4

-2

4

2

-1

1

2ππ

a) b)

c) d)

, ,

, ,

-3

3

2ππ

-2

-1

2

1

-10

10

2ππ

-1

1

2ππ

-2

2

2ππ

Note: All terms from the originalequation were collected on theleft side before graphing.

The graphs are identical.

The graphs are NOT identical.The R.S. has holes.

The graphs are identical.

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a) b)

c) d)

, ,

, ,

-10

10

2ππ

-3

3

2ππ

-2

-1

2

1

1

2ππ

-1

-12

-9

-6

3

02ππ

-3

Note: All terms from the original equation were collected on the left side before graphing.

Note: All terms from the original equation were collected on the left side before graphing.

Example 18:

74

-3

-2

7

a) b) c)

Example 19: See Video

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Answer KeyTrigonometry Lesson Seven: Trigonometric Identities II

Example 1:


Example 2: Example 3:

a) b) c) d) See Video

Example 4: Example 5: See Video

Example 6:

a) i. ii. 0 iii. undefined

b) (answers may vary)

ii.

iii.

iv.

i.

c) (answers may vary)

ii.

iii.

iv.

i.

Examples 7 - 13: Proofs. See Video.

Example 14:

a)

b)

c)

d)

Example 15:

a)

b)

c)

d)

Example 16:

a)

b)

c)

d)

Example 17:

a)

b)

c)

d)

Example 18: 57°

Example 19: 92.9

Example 20:

a)

d

θ

132.2

-132.2

90° 180° 270° 360°

b)

c) θ = 24.6° and θ = 65.4°

Example 21:

a)

b)

c) i. ii. iii.

A

θ

4900

90°45°

At 0°, the cannonball hits the ground as soon as it leaves the cannon, so the horizontal distance is 0 m.

At 45°, the cannonball hits the ground at the maximum horizontal distance, 132.2 m.

At 90°, the cannonball goes straight up and down, landing on the cannon at a horizontal distance of 0 m

The maximum area occurs when θ = 45°. At this angle, the rectangle is the top half of a square.

0

-6

6

2ππ

y = f(θ) + g(θ)

ii. The waves experience constructive interference. iii. The new sound will be louder than either original sound.

0

-6

6

2ππ

y = f(θ) + g(θ)

ii. The waves experience destructive interference.iii. The new sound will be quieter than either original sound.

All of the terms subtract out leaving y = 0, A flat line indicating no wave activity.

Example 22:

a) i. b) i.

c)

Example 23: See Video. Example 24: See Video.

a) b) c) d) e) f)

a) b) c)

a) b) c)

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Answer KeyPermutations and Combinations Lesson One: Permutations

Example 1:a) Six words can be formed. b) 3 × 2 × 1 = 6 c) 3P3 d) See Video e) 3P1 + 3P2 + 3P3Example 2: a) 24 b) 1 c) 1 d) (-2)! Does not exist.e) 20 f) 4 g) n2 – n h) n2 + nExample 3: a) 120 b) 24 c) 720 d) 13!Example 4: a) 3 b) 415800 c) 12600 d) 20 e) 60 f) 10Example 5: a) 8 b) 16 c) 1296 d) 32 × 106 e) 676 000Example 6: a) 120 b) 48 c) 480 d) 108Example 7: a) 24 b) 18 c) 12 d) 18 e) 3 f) 6Example 8: a) 103 680 b) 240 c) 120 d) 144Example 9: a) 72 b) 420 c) 1440Example 10: a) 156 b) 1440 c) 20 d) 144 e) 72Example 11: a) 24 b) 1320 c) 5P2 d) 3P2 or 3P3Example 12: a) n = 6 b) n = 2 c) n = 5 d) n = 1Example 13: a) n = 8 b) r = 3 c) n = 2 d) n = 5

Permutations and Combinations Lesson Two: Combinations

Example 1:a) The order of the colors is not important.b) 6 c) 4C2 d) See Video e) 4C3 + 4C4Example 2: a) 10 b) 126 c) 2598960 d) 36; 84Example 3: a) 13860 b) 720 c) 580008 d) 60Example 4: a) 330 b) 70 c) 1680 d) 19600 e) 13244Example 5: a) 3600 b) 180 c) 75600Example 6: a) 66 b) 84 c) 70 d) 9Example 7: a) 15 b) 20Example 8: a) 81 b) 2594400 c) 2533180 d) 405 e) 31Example 9: a) 21 b) 1 c) 6 d) 6C2 e) 5C1Example 10: a) n = 7 b) 4C2 c) n = 5 d) n = 6Example 11: a) n = 4 b) All n-values c) n = 4 d) n = 4Example 12: a) 6760000 b) 40 c) 1645020 d) 144e) See Video f) 84 g) 504 h) 32Example 13: a) 16!/(2!)8 b) 10 c) 1800 d) 56e) 120 f) 5040 g) 241098 h) 15Example 14: a) 10080 b) 5 c) 8 d) 9e) 92 f) 360 g) 241920 h) 6600Example 15: a) 210 b) 720 c) 5148 d) 131560e) 1024 f) 5 g) 14400 h) 123552Example 16: a) 20 b) See Video c) 100 d) 1152e) n = 8 f) 56 g) See Video h) 36

Permutations and Combinations Lesson Three: The Binomial Theorem

Example 1:a) The eighth row of Pascal's Triangle is: 1, 7, 21, 35, 35, 21, 7, 1.b) See Video. Note that rows and term positions use a zero-based index.c) There is symmetry in each row. For example, the second position of the sixth row is equal to the second-last position of the same row.

Example 2:a) 8C0; 12C10b) 21C2 = 210c) k = 3 and 8, so the fourth and ninth positions have a value of 165.d) 1024

Example 3:a) 20b) 120c) 66d) 54

Example 4:a) The binomial theorem states that a binomial power of the form (x + y)n can be expanded into a series of terms with the form nCkx

n-kyk, where n is the exponent of the binomial (and also the zero-based row of Pascal's Triangle), and k is the zero-based term position.

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Example 5:

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Example 6:

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Example 7:

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Example 8:

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Example 9:

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Example 10:

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Math30-1 Workbook Two

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Transcript of Math30-1 Workbook Two