Chapter Review 541 In Problems 15–29, find the exact value, if any, of each composite function. If there is no value, say it is “not defined.” Do not use a calculator. 15. π ( ) − sin sin 3 8 1 16. π ( ) − cos cos 3 4 1 17. π ( ) − tan tan 2 3 1 18. π ( ) − cos cos 15 7 1 19. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − sin sin 8 9 1 20. ( ) − sin sin 0.9 1 21. ( ) − cos cos 0.6 1 22. [ ] − tan tan 51 23. [ ] ( ) − − cos cos 1.6 1 24. π ( ) − sin cos 2 3 1 25. π ( ) − cos tan 3 4 1 26. − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − tan sin 3 2 1 27. ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ − sec tan 3 3 1 28. ( ) − sin cot 3 4 1 29. ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − tan sin 4 5 1 In Problems 30 and 31, find the inverse function −f 1 of each function f. Find the range of f and the domain and range of −f .1 30. f x x x 2 sin 3 6 6 π π ( ) ( ) = − ≤ ≤ 31. π ( ) = − + ≤ ≤ f x x x cos 3 0 In Problems 32 and 33, write each trigonometric expression as an algebraic expression in u. 32. ( ) − u cos sin 1 33. ( ) − u tan csc 1 In Problems 34–50, establish each identity. 34. θ θ θ θ − = tan cot sin cos 2 2 35. θ θ ( ) + = sin 1 cot 1 2 2 36. θ θ θ + = + 5 cos 3 sin 3 2 cos 2 2 2 37. θ θ θ θ θ − + − = 1 cos sin sin 1 cos 2 csc 38. θ θ θ θ − = − cos cos sin 1 1 tan 39. θ θ θ θ + = − csc 1 csc 1 sin cos2 40. θ θ θ θ − = csc sin cos cot 41. θ θ θ θ − = + 1 sin sec cos 1 sin 3 42. θ θ θ θ θ − = − 1 2 sin sin cos cot tan 2 43. α β α β β α ( ) + = − cos cos sin cot tan 44. cos cos cos 1 tan tan α β α β α β ( ) − = + 45. θ θ θ ( ) + = 1 cos tan 2 sin 46. θ θ θ ( ) = − 2 cot cot 2 cot 1 2 47. θ θ θ ( ) − = 1 8 sin cos cos 4 2 2 48. θ θ θ θ θ ( ) ( ) ( ) − = sin3 cos sin cos 3 sin 2 1 49. θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) + + = sin 2 sin 4 cos 2 cos 4 tan 3 50. cos 2 cos 4 cos 2 cos 4 tan tan 3 0 θ θ θ θ θ θ ( ) ( ) ( ) ( ) ( ) − + − = In Problems 51–58, find the exact value of each expression. 51. ° sin165 52. ° tan105 53. π cos 5 12 54. π ( ) − sin 12 55. ° ° + ° ° cos80 cos20 sin80 sin20 56. ° ° − ° ° sin70 cos40 cos70 sin40 57. π tan 8 58. π sin 5 8 In Problems 59–63, use the information given about the angles α and β to find the exact value of : (a) α β ( ) + sin (b) α β ( ) + cos (c) α β ( ) − sin (d) α β ( ) + tan (e) α ( ) sin 2 (f) β ( ) cos 2 (g) β sin 2 (h) α cos 2 59. α α π β π β π = < < = < < sin 4 5 , 0 2 ; sin 5 13 , 2 60. α π α π β π β π = − < < = < < sin 3 5 , 3 2 ; cos 12 13 , 3 2 2 61. α π α π β β π = < < = < < tan 3 4 , 3 2 ; tan 12 5 , 0 2 62. α π α β π β π = − < < = < < sec 2, 2 0; sec 3, 3 2 2 63. α π α π β π β π = − < < = − < < sin 2 3 , 3 2 ; cos 2 3 , 3 2 In Problems 64–69, find the exact value of each expression. 64. ( ) − − − cos sin 3 5 cos 1 2 1 1 65. ( ) − − − sin cos 5 13 cos 4 5 1 1
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