1. Slide 1-1 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 Chapter 5

2. Slide 1-2 Formulas and Identities Negative Angle Identities

3. Slide 1-3 Formulas and Identities

4. Slide 1-4 Formulas and Identities

5. Slide 1-5 Fundamental Identities 5.1 Fundamental Identities ▪ Using the Fundamental Identities

6. Slide 1-6 If and θ is in quadrant II, find each function value. Example FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (a)sec θ In quadrant II, sec θ is negative, so Pythagorean identity

7. Slide 1-7 (b)sin θ from part (a) Quotient identity Reciprocal identity

8. Slide 1-8 (b)cot(– θ) Reciprocal identity Negative-angle identity

9. Slide 1-9 Write cos x in terms of tan x. Example EXPRESSING ONE FUNCITON IN TERMS OF ANOTHER Since sec x is related to both cos x and tan x by identities, start with Take reciprocals. Reciprocal identity Take the square root of each side. The sign depends on the quadrant of x.

10. Slide 1-10 Write in terms of sin θ and cos θ, and then simplify the expression so that no quotients appear. Quotient identities Multiply numerator and denominator by the LCD. Example

11. Slide 1-11 Reciprocal identity Pythagorean identities Distributive property

12. Slide 1-12 Verifying Trigonometric Identities 5.2 Strategies ▪ Verifying Identities by Working With One Side ▪ Verifying Identities by Working With Both Sides

13. Slide 1-13  As you select substitutions, keep in mind the side you are not changing, because it represents your goal. For example, to verify the identity find an identity that relates tan x to cos x. Since and the secant function is the best link between the two sides.

14. Slide 1-14 Hints for Verifying Identities  If an expression contains 1 + sin x, multiplying both numerator and denominator by 1 – sin x would give 1 – sin 2 x, which could be replaced with cos 2 x. Similar procedures apply for 1 – sin x, 1 + cos x, and 1 – cos x.

15. Slide 1-15 Verifying Identities by Working with One Side To avoid the temptation to use algebraic properties of equations to verify identities, one strategy is to work with only one side and rewrite it to match the other side.

16. Slide 1-16 Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that the following equation is an identity. Work with the right side since it is more complicated. Right side of given equation Distributive property Left side of given equation

17. Slide 1-17 Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that the following equation is an identity. Distributive property Left side Right side

18. Slide 1-18 Example VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity.

19. Slide 1-19 VERIFYING AN IDENTITY (WORKING WITH ONE SIDE) Verify that is an identity. Multiply by 1 in the form Example

20. Slide 1-20 Verifying Identities by Working with Both Sides If both sides of an identity appear to be equally complex, the identity can be verified by working independently on each side until they are changed into a common third result. Each step, on each side, must be reversible.

21. Slide 1-21 Example VERIFYING AN IDENTITY (WORKING WITH BOTH SIDES) Verify that is an identity. Working with the left side: Multiply by 1 in the form Distributive property

22. Slide 1-22 Working with the right side: Factor the numerator. Factor the denominator.

23. Slide 1-23 So, the identity is verified. Left side of given equation Right side of given equation Common third expression

24. Slide 1-24 Sum and Difference Identities for Cosine 5.3 Difference Identity for Cosine ▪ Sum Identity for Cosine ▪ Cofunction Identities ▪ Applying the Sum and Difference Identities ▪ Verifying an Identity

25. Slide 1-25

26. Slide 1-26

27. Slide 1-27

28. Slide 1-28 Example FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 15 .

29. Slide 1-29 Example FINDING EXACT COSINE FUNCTION VALUES Find the exact value of

30. Slide 1-30 Example FINDING EXACT COSINE FUNCTION VALUES Find the exact value of cos 87  cos 93  – sin 87  sin 93 .

31. Slide 1-31 Example USING COFUNCTION IDENTITIES TO FIND θ Find one value of θ or x that satisfies each of the following. (a)cot θ = tan 25° (b)sin θ = cos (–30°)

32. Slide 1-32 Example USING COFUNCTION IDENTITIES TO FIND θ (continued) (c) Find one value of θ or x that satisfies the following.

33. Slide 1-33 Example REDUCING cos (A – B) TO A FUNCTION OF A SINGLE VARIABLE Write cos(180° – θ) as a trigonometric function of θ alone.

34. Slide 1-34 Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t Suppose that and both s and t are in quadrant II. Find cos(s + t). Sketch an angle s in quadrant II such that Since let y = 3 and r =5. The Pythagorean theorem gives Since s is in quadrant II, x = –4 and Method1

35. Slide 1-35 Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.) Sketch an angle t in quadrant II such that Since let x = –12 and r = 13. The Pythagorean theorem gives Since t is in quadrant II, y = 5 and

36. Slide 1-36 Example FINDING cos (s + t) GIVEN INFORMATION ABOUT s AND t (cont.)

37. Slide 1-37 Example Method2 We use Pythagorean identities here. To find cos s, recall that sin 2 s + cos 2 s = 1, where s is in quadrant II. sin s = 3/5 Square. Subtract 9/25 cos s < 0 because s is in quadrant II.

38. Slide 1-38 Example To find sin t, we use sin 2 t + cos 2 t = 1, where t is in quadrant II. cos t = –12/13 Square. Subtract 144/169 sin t > 0 because t is in quadrant II. From this point, the problem is solved using (see Method 1).

39. Slide 1-39 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 ▪ Verifying an Identity

40. Slide 1-40 Sum and Difference Identities for Tangent Fundamental identity Sum identities Multiply numerator and denominator by 1.

41. Slide 1-41 Sum and Difference Identities for Tangent Multiply. Simplify. Fundamental identity Replace B with –B and use the fact that tan(–B) = –tan B to obtain the identity for the tangent of the difference of two angles.

42. Slide 1-42 Tangent of a Sum or Difference

43. Slide 1-43 Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES Find the exact value of sin 75 .

44. Slide 1-44 Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES Find the exact value of

45. Slide 1-45 Example FINDING EXACT SINE AND TANGENT FUNCTION VALUES Find the exact value of

46. Slide 1-46 Example WRITING FUNCTIONS AS EXPRESSIONS INVOLVING FUNCTIONS OF θ Write each function as an expression involving functions of θ. (a) (b) (c)

47. Slide 1-47 Example FINDING FUNCTION VALUES AND THE QUADRANT OF A + B Suppose that A and B are angles in standard position with Find each of the following.

48. Slide 1-48 The identity for sin(A + B) involves sin A, cos A, sin B, and cos B. The identity for tan(A + B) requires tan A and tan B. We must find cos A, tan A, sin B and tan B. Because A is in quadrant II, cos A is negative and tan A is negative.

49. Slide 1-49 Because B is in quadrant III, sin B is negative and tan B is positive.

50. Slide 1-50 (a) (b)

51. Slide 1-51 (c) From parts (a) and (b), sin (A + B) > 0 and tan (A + B) > 0. The only quadrant in which the values of both the sine and the tangent are positive is quadrant I, so (A + B) is in quadrant I.

52. Slide 1-52 Example VERIFYING AN IDENTITY USING SUM AND DIFFERENCE IDENTITIES Verify that the equation is an identity.

53. Slide 1-53 Double-Angle Identities 5.5 Double-Angle Identities ▪ An Application ▪ Product-to-Sum and Sum-to-Product Identities

54. Slide 1-54 Half-Angle Identities 5.6 Half-Angle Identities ▪ Applying the Half-Angle Identities ▪ Verifying an Identity

55. Slide 1-55 Half-Angle Identities We can use the cosine sum identities to derive half-angle identities. Choose the appropriate sign depending on the quadrant of

56. Slide 1-56 Half-Angle Identities Choose the appropriate sign depending on the quadrant of

57. Slide 1-57 Half-Angle Identities There are three alternative forms for

58. Slide 1-58 Half-Angle Identities From the identity we can also derive an equivalent identity.

59. Slide 1-59 Half-Angle Identities

60. Slide 1-60 Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE Find the exact value of cos 15° using the half-angle identity for cosine. Choose the positive square root.

61. Slide 1-61 Example USING A HALF-ANGLE IDENTITY TO FIND AN EXACT VALUE Find the exact value of tan 22.5° using the identity

62. Slide 1-62 Example FINDING FUNCTION VALUES OF s/2 GIVEN INFORMATION ABOUT s The angle associated with lies in quadrant II since is positive while are negative.

63. Slide 1-63

64. Slide 1-64 Example SIMPLIFYING EXPRESSIONS USING THE HALF-ANGLE IDENTITIES Simplify each expression. Substitute 12x for A: This matches part of the identity for

65. Slide 1-65 Example VERIFYING AN IDENTITY Verify that is an identity.

66. Slide 1-66 Double-Angle Identities We can use the cosine sum identity to derive double-angle identities for cosine. Cosine sum identity

67. Slide 1-67 Double-Angle Identities There are two alternate forms of this identity.

68. Slide 1-68 Double-Angle Identities We can use the sine sum identity to derive a double-angle identity for sine. Sine sum identity

69. Slide 1-69 Double-Angle Identities We can use the tangent sum identity to derive a double-angle identity for tangent. Tangent sum identity

70. Slide 1-70 Double-Angle Identities

71. Slide 1-71 Example FINDING FUNCTION VALUES OF 2θ GIVEN INFORMATION ABOUT θ Given and sin θ < 0, find sin 2θ, cos 2θ, and tan 2θ. To find sin 2θ, we must first find the value of sin θ. Now use the double-angle identity for sine. Now find cos2θ, using the first double-angle identity for cosine (any of the three forms may be used).

72. Slide 1-72 Now find tan θ and then use the tangent double- angle identity.

73. Slide 1-73 Alternatively, find tan 2θ by finding the quotient of sin 2θ and cos 2θ.

74. Slide 1-74 Example FINDING FUNCTION VALUES OF θ GIVEN INFORMATION ABOUT 2θ Find the values of the six trigonometric functions of θ if We must obtain a trigonometric function value of θ alone. θ is in quadrant II, so sin θ is positive.

75. Slide 1-75 Use a right triangle in quadrant II to find the values of cos θ and tan θ. Use the Pythagorean theorem to find x.

76. Slide 1-76 Example VERIFYING A DOUBLE-ANGLE IDENTITY Quotient identity Verify that is an identity. Double-angle identity

77. Slide 1-77 Example SIMPLIFYING EXPRESSIONS USING DOUBLE-ANGLE IDENTITIES Simplify each expression. Multiply by 1. cos2A = cos 2 A – sin 2 A

78. Slide 1-78 Example DERIVING A MULTIPLE-ANGLE IDENTITY Write sin 3x in terms of sin x. Sine sum identity Double-angle identities

79. Slide 1-79 Product-to-Sum Identities We can add the identities for cos(A + B) and cos(A – B) to derive a product-to-sum identity for cosines.

80. Slide 1-80 Product-to-Sum Identities Similarly, subtracting cos(A + B) from cos(A – B) gives a product-to-sum identity for sines.

81. Slide 1-81 Product-to-Sum Identities Using the identities for sin(A + B) and sin(A – B) in the same way, we obtain two more identities.

82. Slide 1-82 Product-to-Sum Identities

83. Slide 1-83 Example Write 4 cos 75° sin 25° as the sum or difference of two functions. USING A PRODUCT-TO-SUM IDENTITY

84. Slide 1-84 Sum-to-Product Identities

85. Slide 1-85 Example Write as a product of two functions. USING A SUM-TO-PRODUCT IDENTITY