Basic Terminology
Initial side and terminal side: The rays in an angleAngle shows direction and
amount of rotationLower-case Greek letters
denote angles
Basic Terminology
Positive angle: Counterclockwise rotation
Negative angle: Clockwise rotation
Coterminal angles: Share initial and terminal sides
Positive angle
Positive angle
Negative angle
Basic Terminology
Quadrantal angle: Angle in standard position that doesn’t lie in any quadrant
Lies in quadrant II
Lies in quadrant IV
Quadrantal angle
Measuring Angles
Two usual ways of measuringDegrees
360± in one rotationRadians
2¼ radians in one rotation
Degrees, Minutes and Seconds
One complete revolution = 360±
One minute:One-sixtieth of a degreeOne minute is denoted 10
1± = 600
One second: One-sixtieth of a minuteOne second is denoted 100
10 = 6000
Degrees, Minutes and Seconds
Example. Convert to a decimal in degreesProblem: 64±3502700 Answer:
Example. Convert to the D±M0S00 formProblem: 73.582±
Answer:
Radians
Central angle: An angle whose vertex is at the center of a circleCentral angles subtend an arc
on the circle
Radians
One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle
Radians
IMPORTANT!Radians are dimensionlessIf an angle appears with no
units, it must be assumed to be in radians
Arc Length
Theorem. [Arc Length]For a circle of radius r, a central
angle of µ radians subtends an arc whose length s is
s = rµ
WARNING!The angle must be given in
radians
Arc Length
Example. Problem: Find the length of the
arc of a circle of radius 5 centimeters subtended by a central angle of 1.4 radians
Answer:
Radians vs. Degrees
1 revolution = 2¼ radians = 360±
180± = ¼ radians1± = radians1 radian =
¼180
180¼
±
Radians vs. Degrees
Example. Convert each angle in degrees to radians and each angle in radians to degrees(a) Problem: 45±
Answer: (b) Problem: {270±
Answer: (c) Problem: 2 radians
Answer:
Area of a Sector of a Circle
Theorem. [Area of a Sector]The area A of the sector of a
circle of radius r formed by a central angle of µ radians is
A = 12r2µ
Area of a Sector of a Circle
Example.Problem: Find the area of the
sector of a circle of radius 3 meters formed by an angle of 45±. Round your answer to two decimal places.
Answer:
WARNING!The angle again must be given
in radians
Linear and Angular Speed
Object moving around a circle or radius r at a constant speedLinear speed: Distance traveled
divided by elapsed time
t = timeµ = central angle swept out in time ts = rµ = arc length = distance traveled
v = st
Linear and Angular Speed
Object moving around a circle or radius r at a constant speedAngular speed: Angle swept out
divided by elapsed time
Linear and angular speeds are related
v = r!
! = µt
Linear and Angular Speed
Example. A neighborhood carnival has a Ferris wheel whose radius is 50 feet. You measure the time it takes for one revolution to be 90 seconds.(a) Problem: What is the linear
speed (in feet per second) of this Ferris wheel?
Answer: (b) Problem: What is the angular
speed (in radians per second)? Answer:
Key Points
Basic Terminology Measuring Angles Degrees, Minutes and Seconds Radians Arc Length Radians vs. Degrees Area of a Sector of a Circle Linear and Angular Speed
Unit Circle
Unit circle: Circle with radius 1 centered at the originEquation: x2 + y2 = 1Circumference: 2¼
Unit Circle
Travel t units around circle, starting from the point (1,0), ending at the point P = (x, y)
The point P = (x, y) is used to define the trigonometric functions of t
Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:Sine function: y-coordinate of P
sin t = yCosine function: x-coordinate of
Pcos t = x
Tangent function: if x 0
Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:Cosecant function: if y 0
Secant function: if x 0
Cotangent function: if y 0
Exact Values Using Points on the Circle
A point on the unit circle will satisfy the equation x2 + y2 = 1
Use this information together with the definitions of the trigonometric functions.
Exact Values Using Points on the Circle
Example. Let t be a real number and P = the point on the unit circle that corresponds to t.
Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t
Answer:
Trigonometric Functions of Angles
Convert between arc length and angles on unit circle
Use angle µ to define trigonometric functions of the angle µ
Exact Values for Quadrantal Angles
Quadrantal angles correspond to integer multiples of 90± or of radians
Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µProblem: µ = 0 = 0±
Answer:
Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µProblem: µ = = 90±
Answer:
Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µProblem: µ = ¼ = 180±
Answer:
Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µProblem: µ = = 270±
Answer:
Exact Values for Quadrantal Angles
Example. Find the exact values of(a) Problem: sin({90±)
Answer: (b) Problem: cos(5¼)
Answer:
Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µProblem: µ = = 45±
Answer:
Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µProblem: µ = = 60±
Answer:
Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µProblem: µ = = 30±
Answer:
Exact Values for Standard Angles
Example. Find the values of the following expressions
(a) Problem: sin(315±)
Answer:
(b) Problem: cos({120±)
Answer:
(c) Problem:
Answer:
Approximating Values Using a Calculator
IMPORTANT!Be sure that your calculator is
in the correct mode.
Use the basic trigonometric facts:
Approximating Values Using a Calculator
Example. Use a calculator to find the approximate values of the following. Express your answers rounded to two decimal places.(a) Problem: sin 57±
Answer: (b) Problem: cot {153±
Answer: (c) Problem: sec 2
Answer:
Circles of Radius r
Theorem.For an angle µ in standard
position, let P = (x, y) be the point on the terminal side of µ that is also on the circle x2 + y2 = r2. Then
Circles of Radius r
Example. Problem: Find the exact values
of each of the trigonometric functions of an angle µ if ({12, {5) is a point on its terminal side.
Answer:
Key PointsUnit CircleTrigonometric FunctionsExact Values Using Points on the
Circle Trigonometric Functions of AnglesExact Values for Quadrantal
AnglesExact Values for Standard AnglesApproximating Values Using a
Calculator
Domains of Trigonometric
FunctionsDomain of sine and cosine
functions is the set of all real numbers
Domain of tangent and secant functions is the set of all real numbers, except odd integer multiples of = 90±
Domain of cotangent and cosecant functions is the set of all real numbers, except integer multiples of ¼ = 180±
Ranges of Trigonometric
Functions Sine and cosine have range [{1,
1]{1 · sin µ · 1; jsin µj · 1{1 · cos µ · 1; jcos µj · 1
Range of cosecant and secant is ({1, {1] [ [1, 1) jcsc µj ¸ 1 jsec µj ¸ 1
Range of tangent and cotangent functions is the set of all real numbers
Periods of Trigonometric
FunctionsPeriodic function: A function
f with a positive number p such that whenever µ is in the domain of f, so is µ + p, and
f(µ + p) = f(µ) (Fundamental) period of f:
smallest such number p, if it exists
Periods of Trigonometric
Functions Periodic Properties:
sin(µ + 2¼) = sin µcos(µ + 2¼) = cos µtan(µ + ¼) = tan µcsc(µ + 2¼) = csc µsec(µ + 2¼) = sec µcot(µ + ¼) = cot µ
Sine, cosine, cosecant and secant have period 2¼
Tangent and cotangent have period ¼
Periods of Trigonometric
FunctionsExample. Find the exact values
of
(a) Problem: sin(7¼)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Signs of the Trigonometric
FunctionsP = (x, y) corresponding to
angle µDefinitions of functions, where
defined
Find the signs of the functionsQuadrant I: x > 0, y > 0Quadrant II: x < 0, y > 0Quadrant III: x < 0, y < 0Quadrant IV: x > 0, y < 0
Signs of the Trigonometric
FunctionsExample:
Problem: If sin µ < 0 and cos µ > 0, name the quadrant in which the angle µ lies
Answer:
Quotient Identities
Example.
Problem: Given and
, find the exact values of the
four remaining trigonometric
functions of µ using identities.
Answer:
Pythagorean Identities
Unit circle: x2 + y2 = 1 (sin µ)2 + (cos µ)2 = 1
sin2 µ + cos2 µ = 1tan2 µ + 1 = sec2 µ1 + cot2 µ = csc2 µ
Pythagorean Identities
Example. Find the exact values of each expression. Do not use a calculator
(a) Problem: cos 20± sec 20±
Answer:
(b) Problem: tan2 25± { sec2 25±
Answer:
Pythagorean Identities
Example.
Problem: Given that
and that µ is in Quadrant II,
find cos µ.
Answer:
Even-Odd Properties
A function f is even if f({µ) = f(µ) for all µ in the domain of f
A function f is odd if f({µ) = {f(µ) for all µ in the domain of f
Even-Odd PropertiesTheorem. [Even-Odd Properties]
sin({µ) = {sin(µ)cos({µ) = cos(µ)
tan({µ) = {tan(µ)csc({µ) = {csc(µ)sec({µ) = sec(µ)cot({µ) = {cot(µ)
Cosine and secant are even functions
The other functions are odd functions
Even-Odd Properties
Example. Find the exact values
of
(a) Problem: sin({30±)
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
Fundamental Trigonometric IdentitiesQuotient Identities
Reciprocal Identities
Pythagorean Identities
Even-Odd Identities
Key Points Domains of Trigonometric Functions Ranges of Trigonometric Functions Periods of Trigonometric Functions Signs of the Trigonometric
Functions Quotient Identities Pythagorean Identities Even-Odd Properties Fundamental Trigonometric
Identities
Graphing Trigonometric Functions
Graph in xy-plane Write functions as
y = f(x) = sin x y = f(x) = cos x y = f(x) = tan x y = f(x) = csc x y = f(x) = sec x y = f(x) = cot x
Variable x is an angle, measured in radians Can be any real number
Graphing the Sine Function
Periodicity: Only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Sine Function
Domain: All real numbers
Range: [{1, 1]
Odd function
Periodic, period 2¼
x-intercepts: …, {2¼, {¼, 0, ¼, 2¼, 3¼, …
y-intercept: 0
Maximum value: y = 1, occurring at
Minimum value: y = {1, occurring at
2
32
2 52
32
-4
-2
2
4
Transformations of the Graph of the Sine
FunctionsExample.
Problem: Use the graph of y =
sin x to graph
Answer:
Graphing the Cosine Function
Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph
Properties of the Cosine Function
Domain: All real numbers Range: [{1, 1] Even function Periodic, period 2¼ x-intercepts: y-intercept: 1 Maximum value: y = 1, occurring at
x = …, {2¼, 0, 2¼, 4¼, 6¼, … Minimum value: y = {1, occurring at
x = …, {¼, ¼, 3¼, 5¼, …
2
32
2 52
32
-4
-2
2
4
Example.
Problem: Use the graph of y =
cos x to graph
Answer:
Transformations of the Graph of the Cosine
Functions
Sinusoidal Graphs
Graphs of sine and cosine functions appear to be translations of each other
Graphs are called sinusoidalConjecture.
Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAjNumber jAj is the amplitude
Amplitude and Period of Sinusoidal
FunctionsGraphs of functions y = A sin
x and y = A cos x will always satisfy inequality {jAj · y · jAjNumber jAj is the amplitude
2
32
2 52
32
-4
-2
2
4
2
32
2 52
32
-4
-2
2
4
Amplitude and Period of Sinusoidal Functions
Period of y = sin(!x) and
y = cos(!x) is
2
32
2 52
32
-4
-2
2
4
2
32
2 52
32
-4
-2
2
4
Amplitude and Period of Sinusoidal
FunctionsCycle: One period of y =
sin(!x) or
y = cos(!x)
2
32
2 52
32
-4
-2
2
4
2
32
2 52
32
-4
-2
2
4
Amplitude and Period of Sinusoidal Functions
Theorem. If ! > 0, the
amplitude and period of y =
Asin(!x) and
y = Acos(! x) are given by
Amplitude = j Aj
Period = .
Amplitude and Period of Sinusoidal Functions
Example.Problem: Determine the
amplitude and period of y = {2cos(¼x)
Answer:
Graphing Sinusoidal Functions
One cycle contains four important subintervals
For y = sin x and y = cos x these are
Gives five key points on graph
Finding Equations for Sinusoidal Graphs
Example.
Problem: Find an equation for
the graph.
Answer:
2
322 5
23
2
32
252
3
-6
-4
-2
2
4
6
Key Points
Graphing Trigonometric Functions
Graphing the Sine FunctionProperties of the Sine FunctionTransformations of the Graph of
the Sine FunctionsGraphing the Cosine FunctionProperties of the Cosine FunctionTransformations of the Graph of
the Cosine Functions
Key Points (cont.)
Sinusoidal GraphsAmplitude and Period of
Sinusoidal FunctionsGraphing Sinusoidal
FunctionsFinding Equations for
Sinusoidal Graphs
Graphing the Tangent Function
Periodicity: Only need to graph on interval [0, ¼]
Plot points and graph
Properties of the Tangent Function
Domain: All real numbers, except odd multiples of
Range: All real numbers Odd function Periodic, period ¼ x-intercepts: …, {2¼, {¼, 0, ¼, 2¼,
3¼, … y-intercept: 0 Asymptotes occur at
2
32
2 52
32
-8
-6
-4
-2
2
4
6
8
Transformations of the Graph of the Tangent
FunctionsExample.
Problem: Use the graph of y =
tan x to graph
Answer:
Key Points
Graphing the Tangent FunctionProperties of the Tangent
FunctionTransformations of the Graph
of the Tangent FunctionsGraphing the Cotangent
FunctionGraphing the Cosecant and
Secant Functions
Graphing Sinusoidal Functions
y = A sin(!x), ! > 0Amplitude jAjPeriod
y = A sin(!x { Á)Phase shiftPhase shift indicates amount of
shift To right if Á > 0
To left if Á < 0
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á):
Determine amplitude jAj
Determine period
Determine starting point of one
cycle:
Determine ending point of one
cycle:
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or y = A cos(!x { Á):
Divide interval into
four subintervals, each with
length
Use endpoints of subintervals to
find the five key points
Fill in one cycle
Graphing Sinusoidal Functions
Graphing y = A sin(!x { Á) or
y = A cos(!x { Á):
Extend the graph in each
direction to make it complete
Graphing Sinusoidal Functions
Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
Answer:
(c) Problem: Find the phase shift
Answer:
Finding a Sinusoidal Function from Data
Example. An experiment in a wind tunnel generates cyclic waves. The following data is collected for 52 seconds.Let v represent the wind speed in feet per second and let x represent the time in seconds.
Time (in seconds), x Wind speed (in feet per second), v
0 21
12 42
26 67
41 40
52 20
Finding a Sinusoidal Function from Data
Example. (cont.)Problem: Write a sine equation
that represents the dataAnswer:
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