1. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-1 5.1 Fundamental Identities 5.2 Verifying Trigonometric Identities 5.3 Sum and Difference Identities for Cosine 5.4 Sum and Difference Identities for Sine and Tangent 5.5Double-Angle Identities 5.6Half-Angle Identities Trigonometric Identities 5

2. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-2 Trigonometric Identities 5.1 Fundamental Identities ▪ Using the Fundamental Identities

3. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-3 If and is in quadrant IV, find each function value. 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (a) In quadrant IV, is negative.

4. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-4 If and is in quadrant IV, find each function value. 5.1 Example 1 Finding Trigonometric Function Values Given One Value and the Quadrant (cont.) (b) (c)

5. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-5 5.1 Example 2 Expressing One Function in Terms of Another

6. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-6 5.1 Example 3 Rewriting an Expression in Terms of Sine and Cosine Write in terms of and, and then simplify the expression.

7. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-7 Verifying Trigonometric Identities 5.2 Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides

8. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-8 Verify that is an identity. 5.2 Example 1 Verifying an Identity (Working With One Side) Left side of given equation Right side of given equation

9. 5-9 Verify that is an identity. 5.2 Example 2 Verifying an Identity (Working With One Side) Simplify.

10. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-10 Verify that is an identity. 5.2 Example 3 Verifying an Identity (Working With One Side) Factor. Simplify.

11. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-11 Verify that is an identity. 5.2 Example 4 Verifying an Identity (Working With One Side) Multiply by 1 in the form

12. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-12 Verify that is an identity. 5.2 Example 5 Verifying an Identity (Working With Both Sides) Working with the left side: Multiply by 1 in the form Simplify.

13. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-13 5.2 Example 5 Verifying an Identity (Working With Both Sides) (cont.) Working with the right side: Factor the numerator. Distributive property. Factor the numerator. Factor the denominator. Simplify.

14. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-14 5.2 Example 5 Verifying an Identity (Working With Both Sides) (cont.) So, the identity is verified.

15. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-15 Sum and Difference Identities for Cosine 5.3 Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities

16. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-16 Find the exact value of each expression. 5.3 Example 1 Finding Exact Cosine Function Values (a)

17. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-17 5.3 Example 1 Finding Exact Cosine Function Values (cont.) (b) (c)

18. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-18 5.3 Example 2 Using Cofunction Identities to Find θ Find an angle θ that satisfies each of the following.

19. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-19 5.3 Example 3 Reducing cos ( A – B ) to a Function of a Single Variable Write cos(90° + θ) as a trigonometric function of θ alone.

20. 5-20 5.3 Example 2B Using Cofunction Identities to Find θ ( Miscellaneous HW Examples ) Find an angle θ that satisfies each of the following. 1.Sin (θ + 15 o ) = Cos (2θ + 5 o ) Now see 5.3 # 38 2.Write Cos π/12 as cofunction Now see 5.3 #18

21. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-21 5.3 Example 4 Finding cos ( s + t ) Given Information About s and t Suppose that, and both s and t are in quadrant IV. Find cos(s – t). The Pythagorean theorem gives Since s is in quadrant IV, y = –8.

22. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-22 5.3 Example 4 Finding cos ( s + t ) Given Information About s and t (cont.) Use a Pythagorean identity to find the value of cos t. Since t is in quadrant IV,

23. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-23 5.3 Example 4 Finding cos ( s + t ) Given Information About s and t (cont.)

24. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-24 Sum and Difference Identities for Sine and Tangent 5.4 Sum and Difference Identities for Sine ▪ Sum and Difference Identities for Tangent ▪ Applying the Sum and Difference Identities

25. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-25 Find the exact value of each expression. 5.4 Example 1 Finding Exact Sine and Tangent Function Values (a)

26. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-26 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (b)

27. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-27 5.4 Example 1 Finding Exact Sine and Tangent Function Values (cont.) (c)

28. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-28 Write each function as an expression involving functions of θ. 5.4 Example 2 Writing Functions as Expressions Involving Functions of θ (a) (b) (c)

29. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-29 5.4 Example 4 Verifying an Identity Using Sum and Difference Identities Verify that the equation is an identity. Combine the fractions.

30. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-30 5.4 Example 4 Verifying an Identity Using Sum and Difference Identities (cont.) Expand the terms. Combine terms. Factor. So, the identity is verified.

31. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-31 Double-Angle Identities 5.5 Double-Angle Identities ▪ Omit Product-to-Sum and Sum-to- Product Identities

32. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-32 5.5 Example 1 Finding Function Values of 2 θ Given Information About θ The identity for sin 2θ requires cos θ.

33. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-33 5.5 Example 1 Finding Function Values of 2 θ Given Information About θ (cont.)

34. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-34 5.5 Example 1 Finding Function Values of 2 θ Given Information About θ (cont.) Alternatively, find tan θ and then use the tangent double-angle identity.

35. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-35 5.5 Example 2 Finding Function Values of θ Given Information About 2 θ Find the values of the six trigonometric functions of θ if Use the identity to find sin θ: θ is in quadrant III, so sin θ is negative.

36. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-36 5.5 Example 2 Finding Function Values of θ Given Information About 2 θ (cont.) Use the identity to find cos θ: θ is in quadrant III, so cos θ is negative.

37. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-37 5.5 Example 2 Finding Function Values of θ Given Information About 2 θ (cont.)

38. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-38 Half-Angle Identities 5.6 Half-Angle Identities ▪ Applying the Half-Angle Identities

39. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-39 5.6 Example 1 Using a Half-Angle Identity to Find an Exact Value Find the exact value of sin 22.5° using the half-angle identity for sine.

40. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-40 5.6 Example 2 Using a Half-Angle Identity to Find an Exact Value Find the exact value of tan 75° using the identity

41. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-41 The angle associated with lies in quadrant II since is positive while are negative. 5.6 Example 3 Finding Function Values of Given Information About s s2s2

42. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 5-42 5.6 Example 3 Finding Function Values of Given Information About s (cont.) s2s2

43. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-43 6.1 Inverse Circular Functions 6.2 Trigonometric Equations I 6.3 Trigonometric Equations II 6.4 Equations Involving Inverse Trigonometric Functions Inverse Circular Functions and Trigonometric Equations 6

44. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-44 Inverse Circular Functions 6.1 Inverse Functions ▪ Inverse Sine Function ▪ Inverse Cosine Function ▪ Inverse Tangent Function ▪ Remaining Inverse Circular Functions ▪ Inverse Function Values

45. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-45 Find y in each equation. 6.1 Example 1 Finding Inverse Sine Values

46. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-46 6.1 Example 1 Finding Inverse Sine Values (cont.)

47. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-47 6.1 Example 1 Finding Inverse Sine Values (cont.) is not in the domain of the inverse sine function, [–1, 1], so does not exist. A graphing calculator will give an error message for this input.

48. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-48 Find y in each equation. 6.1 Example 2 Finding Inverse Cosine Values

49. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-49 Find y in each equation. 6.1 Example 2 Finding Inverse Cosine Values

50. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-50 6.1 Example 3 Finding Inverse Function Values (Degree- Measured Angles) Find the degree measure of θ in each of the following.

51. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-51 6.1 Example 4 Finding Inverse Function Values With a Calculator (a)Find y in radians if With the calculator in radian mode, enter as y = 1.823476582

52. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-52 6.1 Example 4(b) Finding Inverse Function Values With a Calculator (b)Find θ in degrees if θ = arccot(–.2528). A calculator gives the inverse cotangent value of a negative number as a quadrant IV angle. The restriction on the range of arccotangent implies that the angle must be in quadrant II, so, with the calculator in degree mode, enter arccot(–.2528) as

53. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-53 6.1 Example 4(b) Finding Inverse Function Values With a Calculator (cont.) θ = 104.1871349°

54. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-54 6.1 Example 5 Finding Function Values Using Definitions of the Trigonometric Functions Evaluate each expression without a calculator. Since arcsin is defined only in quadrants I and IV, and is positive, θ is in quadrant I.

55. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-55 6.1 Example 5(a) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

56. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-56 6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions Since arccot is defined only in quadrants I and II, and is negative, θ is in quadrant II.

57. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-57 6.1 Example 5(b) Finding Function Values Using Definitions of the Trigonometric Functions (cont.)

58. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-58 6.1 Example 6(a) Finding Function Values Using Identities Evaluate the expression without a calculator. Use the cosine difference identity:

59. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-59 6.1 Example 6(a) Finding Function Values Using Identities (cont.) Sketch both A and B in quadrant I. Use the Pythagorean theorem to find the missing side.

60. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-60 6.1 Example 6(a) Finding Function Values Using Identities (cont.)

61. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-61 6.1 Example 6(b) Finding Function Values Using Identities Evaluate the expression without a calculator. Use the double-angle sine identity: sin(2 arccot (–5)) Let A = arccot (–5), so cot A = –5.

62. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-62 6.1 Example 6(b) Finding Function Values Using Identities (cont.) Sketch A in quadrant II. Use the Pythagorean theorem to find the missing side.

63. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-63 6.1 Example 6(b) Finding Function Values Using Identities (cont.)

64. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-64 Trigonometric Equations I 6.2 Solving by Linear Methods ▪ Solving by Factoring ▪ Solving by Quadratic Methods ▪ Solving by Using Trigonometric Identities

65. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-65 6.2 Example 1 Solving a Trigonometric Equation by Linear Methods is positive in quadrants I and III. The reference angle is 30° because

66. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-66 6.2 Example 1 Solving a Trigonometric Equation by Linear Methods (cont.) Solution set: {30°, 210°}

67. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-67 6.2 Example 2 Solving a Trigonometric Equation by Factoring or Solution set: {90°, 135°, 270°, 315°}

68. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-68 6.2 Example 3 Solving a Trigonometric Equation by Factoring

69. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-69 6.2 Example 3 Solving a Trigonometric Equation by Factoring (cont.) has one solution, has two solutions, the angles in quadrants III and IV with the reference angle.729728: 3.8713 and 5.5535.

70. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-70 Trigonometric Equations II 6.3 Equations with Half-Angles ▪ Equations with Multiple Angles

71. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-71 6.3 Example 1 Solving an Equation Using a Half-Angle Identity (a)over the interval and (b)give all solutions. is not in the requested domain.

72. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-72 6.3 Example 2 Solving an Equation With a Double Angle Factor. or

73. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-73 6.3 Example 3 Solving an Equation Using a Multiple Angle Identity From the given interval 0 ° ≤ θ < 360°, the interval for 2θ is 0 ° ≤ 2θ < 720°.

74. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-74 6.3 Example 3 Solving an Equation Using a Multiple Angle Identity (cont.) Since cosine is negative in quadrants II and III, solutions over this interval are

75. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-75 Equations Involving Inverse Trigonometric Functions 6.4 Solving for x in Terms of y Using Inverse Functions ▪ Solving Inverse Trigonometric Equations

76. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-76 6.4 Example 1 Solving an Equation for a Variable Using Inverse Notation

77. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-77 6.4 Example 2 Solving an Equation Involving an Inverse Trigonometric Function

78. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-78 6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions

79. Copyright © 2008 Pearson Addison-Wesley. All rights reserved. 6-79 6.4 Example 3 Solving an Equation Involving Inverse Trigonometric Functions (cont.) Sketch u in quadrant I. Use the Pythagorean theorem to find the missing side.