48. Error Analysis A student tries to show that sin ( A + B ) = sin A + sin B sine open eh plus b close equals sine eh plus sine b is true by letting A = 120 ° eh equals , 120 degrees and B = 240 ° . b equals 240 degrees . Why is the student's reasoning not correct?

    

C Challenge

49. 1. Reasoning A function is even if f ( − x ) = f ( x ) . f open negative x close equals f open x close . A function is odd if f ( − x ) = − f ( x ) . f open negative x close equals negative f open x close . Which trigonometric functions are even? Which are odd?
    2. Writing Are all functions either even or odd? Explain your answer. Give a counterexample if possible.

Use the sum and difference formulas to verify each identity.

50. cos ( π − θ ) = − cos θ cosine open pi negative theta close equals negative cosine theta
51. sin ( π − θ ) = sin θ sine open pi negative theta close equals sine theta
52. sin ( π + θ ) = − sin θ sine open pi plus theta close equals negative sine theta
53. cos ( π + θ ) = − cos θ cosine open pi plus theta close equals negative cosine theta
54. sin ( 3 π 2 − θ ) = − cos θ sine . open . fraction 3 pi , over 2 end fraction , minus theta . close . equals negative cosine theta
55. cos ( θ + 3 π 2 ) = sin θ cosine . open . theta plus , fraction 3 pi , over 2 end fraction . close . equals sine theta

Standardized Test Prep

SAT/ACT

56. Which expressions are equivalent?
    1. − tan ( π 2 − θ ) negative tangent . open . pi over 2 , minus theta . close
    2. tan ( θ − π 2 ) tangent . open . theta minus , pi over 2 . close
    3. tan ( − ( π 2 − θ ) ) tangent . open . negative . open . pi over 2 , minus theta . close . close
       1. I and II only
       2. II and III only
       3. I and III only
       4. I, II, and III
57. Which expression is equal to cos 50 ° ? cosine , 50 degrees question mark
    6. sin 20 ° cos 30 ° + cos 20 ° sin 30 ° sine , 20 degrees cosine , 30 degrees plus cosine , 20 degrees sine , 30 degrees
    7. cos 20 ° cos 30 ° + sin 20 ° sin 30 ° cosine , 20 degrees cosine , 30 degrees plus sine , 20 degrees sine , 30 degrees
    8. sin 20 ° cos 30 ° − cos 20 ° sin 30 ° sine , 20 degrees cosine , 30 degrees negative cosine , 20 degrees sine , 30 degrees
    9. cos 20 ° cos 30 ° − sin 20 ° sin 30 ° cosine , 20 degrees cosine , 30 degrees negative sine , 20 degrees sine , 30 degrees
58. Which expression is NOT equivalent to cos θ ? cosine theta question mark
    1. − sin ( θ − 90 ° ) negative sine open theta negative 90 degrees close
    2. − cos ( − θ ) negative cosine open negative theta close
    3. sin ( θ + 90 ° ) sine open theta plus 90 degrees close
    4. − cos ( θ + 180 ° ) negative cosine open theta plus 180 degrees close

Short Response

59. Find an exact value for sin 165 ° . sine , 165 degrees . Show your work.

Mixed Review

Use the Law of Cosines. Find the indicated length to the nearest tenth. See Lesson 14-5.

60. In Δ DEF , m ∠ E = 54 ° , d = 14 cap delta ft, and f = 20 ft. f equals 20 , ft. Find e.
61. In Δ RST , m ∠ T = 32 ° , r = 10 c m , cap delta and s = 17 c m . s equals 17 c m . Find t.

Write each measure in radians. Express the answer in terms of π pi and as a decimal rounded to the nearest hundredth. See Lesson 13-3.

62. 80 ° 80 degrees
63. − 50 ° negative 50 degrees
64. − 15 ° negative 15 degrees
65. 70 ° 70 degrees
66. 190 ° 190 degrees

Get Ready! To prepare for Lesson 14-7, do Exercises 67–69.

Complete the following angle identities. See Lesson 14-6.

67. cos ( A + B ) = □ cosine open eh plus b close equals white square
68. sin ( A + B ) = □ sine open eh plus b close equals white square
69. tan ( A + B ) = □ tangent open eh plus b close equals white square

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