1. Trigonometric Functions
Chapter 5
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2. Angles and Their Measure
Section 5.1

3. Basic Terminology Ray: A half-line starting at a vertex V
Angle: Two rays with a common vertex

4. Basic Terminology Initial side and terminal side: The rays in an angle
Angle shows direction and amount of rotation
Lower-case Greek letters denote angles

5. Basic Terminology Positive angle: Counterclockwise rotation
Negative angle: Clockwise rotation
Coterminal angles: Share initial and terminal sides
Positive angle
Negative angle
Positive angle

6. Basic Terminology Standard position: Vertex at origin
Initial side is positive x-axis

7. Basic Terminology
Quadrantal angle: Angle in standard position that doesn’t lie in any quadrant
Lies in quadrant II
Quadrantal angle
Lies in quadrant IV

8. Measuring Angles Two usual ways of measuring Degrees Radians
360± in one rotation
Radians
2¼ radians in one rotation

9. Measuring Angles Right angle: A quarter revolution
A right angle contains
90±
radians

10. Measuring Angles Straight angle: A half revolution.
A straight angle contains:
180±
¼ radians

11. Measuring Angles Negative angles have negative measure
Multiple revolutions are allowed

12. Degrees, Minutes and Seconds
One complete revolution = 360±
One minute:
One-sixtieth of a degree
One minute is denoted 10
1± = 600
One second:
One-sixtieth of a minute
One second is denoted 100
10 = 6000

13. Degrees, Minutes and Seconds
Example. Convert to a decimal in degrees
Problem: 64±
Answer:
Example. Convert to the D±M0S00 form
Problem: ±

14. Radians
Central angle: An angle whose vertex is at the center of a circle
Central angles subtend an arc on the circle

15. Radians
One radian is the measure of an angle which subtends an arc with length equal to the radius of the circle

16. Radians IMPORTANT! Radians are dimensionless
If an angle appears with no units, it must be assumed to be in radians

17. Arc Length Theorem. [Arc Length] WARNING!
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

18. 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:

19. Radians vs. Degrees 1 revolution = 2¼ radians = 360± 1± = radians

20. 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±
(c) Problem: 2 radians

21. Radians vs. Degrees
Measurements of common angles

22. 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

23. 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

24. Linear and Angular Speed
Object moving around a circle or radius r at a constant speed
Linear speed: Distance traveled divided by elapsed time
t = time
µ = central angle swept out in time t
s = rµ = arc length = distance traveled

25. Linear and Angular Speed
Object moving around a circle or radius r at a constant speed
Angular speed: Angle swept out divided by elapsed time
Linear and angular speeds are related
v = r!

26. 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)?

27. 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

28. Trigonometric Functions: Unit Circle Approach
Section 5.2

29. Unit Circle Unit circle: Circle with radius 1 centered at the origin
Equation: x2 + y2 = 1
Circumference: 2¼

30. 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

31. 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 = y
Cosine function: x-coordinate of P
cos t = x
Tangent function: if x  0

32. 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

33. 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.

34. 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:

35. Trigonometric Functions of Angles
Convert between arc length and angles on unit circle
Use angle µ to define trigonometric functions of the angle µ

36. Exact Values for Quadrantal Angles
Quadrantal angles correspond to integer multiples of 90± or of radians

37. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = 0 = 0±
Answer:

38. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = = 90±
Answer:

39. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = ¼ = 180±
Answer:

40. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = = 270±
Answer:

41. Exact Values for Quadrantal Angles

42. Exact Values for Quadrantal Angles
Example. Find the exact values of
(a) Problem: sin({90±)
Answer:
(b) Problem: cos(5¼)

43. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = = 45±
Answer:

44. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = = 60±
Answer:

45. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of µ
Problem: µ = = 30±
Answer:

46. Exact Values for Standard Angles

47. Exact Values for Standard Angles
Example. Find the values of the following expressions
(a) Problem: sin(315±)
Answer:
(b) Problem: cos({120±)
(c) Problem:

48. Approximating Values Using a Calculator
IMPORTANT!
Be sure that your calculator is in the correct mode.
Use the basic trigonometric facts:

49. 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±
(c) Problem: sec 2

50. 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

51. 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:

52. Key Points Unit Circle Trigonometric Functions
Exact Values Using Points on the Circle
Trigonometric Functions of Angles
Exact Values for Quadrantal Angles
Exact Values for Standard Angles
Approximating Values Using a Calculator

53. Key Points (cont.)
Circles of Radius r

54. Properties of the Trigonometric Functions
Section 5.3

55. Domains of Trigonometric Functions
Domain 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±

56. 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

57. Periods of Trigonometric Functions
Periodic 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

58. Periods of Trigonometric Functions
Periodic Properties:
sin(µ + 2¼) = sin µ
cos(µ + 2¼) = cos µ
tan(µ + ¼) = tan µ
csc(µ + 2¼) = csc µ
sec(µ + 2¼) = sec µ
cot(µ + ¼) = cot µ
Sine, cosine, cosecant and secant have period 2¼
Tangent and cotangent have period ¼

59. Periods of Trigonometric Functions
Example. Find the exact values of
(a) Problem: sin(7¼)
Answer:
(b) Problem:
(c) Problem:

60. Signs of the Trigonometric Functions
P = (x, y) corresponding to angle µ
Definitions of functions, where defined
Find the signs of the functions
Quadrant I: x > 0, y > 0
Quadrant II: x < 0, y > 0
Quadrant III: x < 0, y < 0
Quadrant IV: x > 0, y < 0

61. Signs of the Trigonometric Functions

62. Signs of the Trigonometric Functions
Example:
Problem: If sin µ < 0 and cos µ > 0, name the quadrant in which the angle µ lies
Answer:

63. Quotient Identities P = (x, y) corresponding to angle µ:
Get quotient identities:

64. Quotient Identities Example.
Problem: Given and , find the exact values of the four remaining trigonometric functions of µ using identities.
Answer:

65. Pythagorean Identities
Unit circle: x2 + y2 = 1
(sin µ)2 + (cos µ)2 = 1
sin2 µ + cos2 µ = 1
tan2 µ + 1 = sec2 µ
1 + cot2 µ = csc2 µ

66. 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±

67. Pythagorean Identities
Example.
Problem: Given that and that µ is in Quadrant II, find cos µ.
Answer:

68. 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

69. Even-Odd Properties Theorem. [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

70. Even-Odd Properties Example. Find the exact values of
(a) Problem: sin({30±)
Answer:
(b) Problem:
(c) Problem:

71. Fundamental Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
Even-Odd Identities

72. 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

73. Graphs of the Sine and Cosine Functions
Section 5.4

74. 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

75. Graphing the Sine Function
Periodicity: Only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph

76. 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

77. Transformations of the Graph of the Sine Functions
Example.
Problem: Use the graph of y = sin x to graph
Answer:

78. Graphing the Cosine Function
Periodicity: Again, only need to graph on interval [0, 2¼] (One cycle)
Plot points and graph

79. 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, …

80. Transformations of the Graph of the Cosine Functions
Example.
Problem: Use the graph of y = cos x to graph
Answer:

81. Sinusoidal Graphs
Graphs of sine and cosine functions appear to be translations of each other
Graphs are called sinusoidal
Conjecture.

82. Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj
Number jAj is the amplitude

83. Amplitude and Period of Sinusoidal Functions
Graphs of functions y = A sin x and y = A cos x will always satisfy inequality {jAj · y · jAj
Number jAj is the amplitude

84. Amplitude and Period of Sinusoidal Functions
Period of y = sin(!x) and y = cos(!x) is

85. Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

86. Amplitude and Period of Sinusoidal Functions
Cycle: One period of y = sin(!x) or y = cos(!x)

87. 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 =

88. Amplitude and Period of Sinusoidal Functions
Example.
Problem: Determine the amplitude and period of y = {2cos(¼x)
Answer:

89. 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

90. Graphing Sinusoidal Functions
Example.
Problem: Graph y = {3cos(2x)
Answer:

91. Finding Equations for Sinusoidal Graphs
Example.
Problem: Find an equation for the graph.
Answer:

92. Key Points Graphing Trigonometric Functions Graphing the Sine Function
Properties of the Sine Function
Transformations of the Graph of the Sine Functions
Graphing the Cosine Function
Properties of the Cosine Function
Transformations of the Graph of the Cosine Functions

93. Key Points (cont.) Sinusoidal Graphs
Amplitude and Period of Sinusoidal Functions
Graphing Sinusoidal Functions
Finding Equations for Sinusoidal Graphs

94. Graphs of the Tangent, Cotangent, Cosecant and Secant Functions
Section 5.5

95. Graphing the Tangent Function
Periodicity: Only need to graph on interval [0, ¼]
Plot points and graph

96. 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

97. Transformations of the Graph of the Tangent Functions
Example.
Problem: Use the graph of y = tan x to graph
Answer:

98. Graphing the Cotangent Function
Periodicity: Only need to graph on interval [0, ¼]

99. Graphing the Cosecant and Secant Functions
Use reciprocal identities
Graph of y = csc x

100. Graphing the Cosecant and Secant Functions
Use reciprocal identities
Graph of y = sec x

101. Key Points Graphing the Tangent Function
Properties of the Tangent Function
Transformations of the Graph of the Tangent Functions
Graphing the Cotangent Function
Graphing the Cosecant and Secant Functions

102. Phase Shifts; Sinusoidal Curve Fitting
Section 5.6

103. Graphing Sinusoidal Functions
y = A sin(!x), ! > 0
Amplitude jAj
Period
y = A sin(!x { Á)
Phase shift
Phase shift indicates amount of shift
To right if Á > 0
To left if Á < 0

104. 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:

105. 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

106. 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

107. Graphing Sinusoidal Functions
Example. For the equation
(a) Problem: Find the amplitude
Answer:
(b) Problem: Find the period
(c) Problem: Find the phase shift

108. 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 21 12 42 26 67 41 40 52 20

109. Finding a Sinusoidal Function from Data
Example. (cont.)
Problem: Write a sine equation that represents the data
Answer:

110. Key Points Graphing Sinusoidal Functions
Finding a Sinusoidal Function from Data