1. Chapter 5
Sections 2.3 – 2.4 – 2.5

2. Y = SIN X
X is your Angle (θ)
Y is your Height

3. Table

4. y = Sin(x) < x < 2π

5. y = sin x,

6. Definition

7. Y = Sin x Sin(-x) = - Sin(x)
ODD FUNCTION: Symmetric About Origin Like f(x) = x3
Sin(-x) = - Sin(x)
Example:
Sin(- 𝜋 2 )=−1=−𝑆𝑖𝑛( 𝜋 2 )

8. Y = Sin x Sin(-2) = Sin(0) = Sin(2) = Sin(2n), for any Integer n
Period = 2𝜋
Sin(-2) = Sin(0) = Sin(2) = Sin(2n), for any Integer n
Sin(θ) = Sin(θ + 2n) , for any Integer n
FUNDAMENTAL PERIOD: First Complete Cycle (2𝜋)
Amplitute: Max Absolute Value of Height = 1

10. Graph of y= csc x = 1 𝑆𝑖𝑛 𝑥
Domain: ALL REALS, Except Integer Multiples of 
Range: (-∞, -1] U [1, ∞)
NO AMPLITUTE Period = 2

11. Y = COS X
X is your Angle (θ)
Y is your Height

12. Table

13. y = cos x, < x < 2π

14. Figure: y = cos x,

15. Y = Cos x Cos(-x) = Cos(x)
EVEN FUNCTION: Symmetric About Y-Axis Like f(x) = x2
Cos(-x) = Cos(x)
Example:
Cos(- 𝜋 2 )=0=Cos( 𝜋 2 )

16. Y = Cos x Cos(-2) = Cos(0) = Cos(2) = Cos(2n), for any Integer n
Period = 2𝜋
Cos(-2) = Cos(0) = Cos(2) = Cos(2n), for any Integer n
Cos(θ) = Cos(θ + 2n) , for any Integer n
FUNDAMENTAL PERIOD: First Complete Cycle (2𝝅)
Amplitute: Max Absolute Value of Height = 1

18. Graph of y= sec x = 1 𝐶𝑜𝑠 𝑥
Domain: ALL REALS, Except Odd Integer Multiples of 𝜋 2
Range: (-∞, -1] U [1, ∞)
NO AMPLITUTE Period = 2

19. Y = Tan X = 𝑆𝑖𝑛 𝑋 𝐶𝑜𝑠 𝑋
X is your Angle (θ)
Y is your Height

20. Graph of

21. Table

22. Figure: y = tan x

23. Y = Tan x Tan(-x) = - Tan(x)
ODD FUNCTION: Symmetric About Origin Like f(x) = x3
Tan(-x) = - Tan(x)
Example:
Tan(- 𝜋 4 )=−1=−Tan( 𝜋 4 )

24. Y = Tan x Tan(-) = Tan(0) = Tan() = Tan(n), for any Integer n
Period = 𝜋
Tan(-) = Tan(0) = Tan() = Tan(n), for any Integer n
Tan(θ) = Tan(θ + n) , for any Integer n
FUNDAMENTAL PERIOD: First Complete Cycle (𝜋)
Amplitute: NONE Domain: All Real Numbers, Except Odd Multiples of 𝜋 2 Range: (-∞, ∞)

26. Table: y = cot x

27. Figure: Graph of y = cot x

28. Y = Cot x = 1 𝑇𝑎𝑛 𝑋 Cot(-x) = - Cot(x)
ODD FUNCTION: Symmetric About Origin Like f(x) = x3
Cot(-x) = - Cot(x)
Example:
Cot(- 𝜋 4 )=−1=−Cot( 𝜋 4 )

29. Y = Cot x
Period = 𝜋
Cot(-3/2) = Cot(-/2) = Tan(/2) = Tan(n/2), for any Integer n
Cot(θ) = Cot(θ + n) , for any Integer n
FUNDAMENTAL PERIOD: First Complete Cycle (𝜋)
Amplitute: NONE Domain: All Real Numbers, Except Odd Multiples of  Range: (-∞, ∞)

30. Table

31. Theorem

32. Graph Functions of the Form y=A sin(wx) Using Transformations

33. Solution

34. Figure: y = - sin(2x)

35. Graph Functions of the Form y=A cos(wx) Using Transformations

36. Example

37. Solution

38. Figure

40. Example

41. Solution

42. Find an Equation for a Sinusoidal Graph

43. Figure

44. Example
Figure 94

45. Solution

46. Graph Functions of the Form y=A tan(wx)+B and y=A cot(wx)+B

47. Example

48. Solution
Figure 97

49. Example

50. Solution

51. Solution continued

52. Graph Functions of the Form y=A csc(wx)+B and y=A sec(wx)+B

53. Example

54. Solution
Figure 102

55. Example

56. Solution

57. Figure

58. Determine the Signs of the Trigonometric Functions in a Quadrant

59. Figure

60. Table

61. Figure

62. Example

63. Solution

64. Use Even-Odd Properties to Find the Exact Values of the Trigonometric Functions

66. Figure

67. Example

68. Solution

69. Find the Values of the Trigonometric Functions Using Fundamental Identities

73. Example

74. Solution

75. Example

76. Solution

77. Find the Exact Values of the Trigonometric Functions of an Angle Given One of the Functions and the Quadrant of the Angle

78. Example

79. Solution Option 1 Using a Circle

80. Solution Option 1 Using a Circle continued

81. Solution Option 2 Using Identities

82. Solution Option 2 Using Identities continued

84. Example

85. Solution Option 1 Using a Circle

86. Figure

87. Solution Option 2 Using Identities

88. Solution Option 2 Using Identities continued