424 CHAPTER 6 Trigonometric Functions In Problems 35–42, θ sin and θ cos are given. Find the exact value of each of the four remaining trigonometric functions. 35. θ θ = − = sin 3 5 , cos 4 5 36. θ θ = = − sin 4 5 , cos 3 5 37. θ θ = = sin 2 5 5 , cos 5 5 38. θ θ = − = − sin 5 5 , cos 2 5 5 39. θ θ = = sin 1 2 , cos 3 2 40. θ θ = = sin 3 2 , cos 1 2 41. θ θ = − = sin 1 3 , cos 2 2 3 42. θ θ = = − sin 2 2 3 , cos 1 3 In Problems 43–58, find the exact value of each of the remaining trigonometric functions of θ. 4 3. θ θ = sin 12 13 , in quadrant II 44. θ θ = cos 3 5 , in quadrant IV 45. θ θ = − cos 4 5 , in quadrant III 46. θ θ = − sin 5 13 , in quadrant III 47. θ θ = ° < < ° sin 5 13 , 90 180 48. θ θ = ° < < ° cos 4 5 , 270 360 49. θ π θ π = − < < cos 1 3 , 2 50. θ π θ π = − < < sin 2 3 , 3 2 51. θ θ = < sin 2 3 , tan 0 52. θ θ = − > cos 1 4 , tan 0 53. θ θ = < sec 2, sin 0 54. θ θ = < csc 3, cot 0 55. θ θ = < tan 3 4 , sin 0 56. θ θ = < cot 4 3 , cos 0 57. θ θ = − > tan 1 3 , sin 0 58. θ θ = − > sec 2, tan 0 In Problems 59–76, use the even-odd properties to find the exact value of each expression. Do not use a calculator. 59. ( ) − ° sin 60 60. ( ) − ° cos 30 61. ( ) − ° tan 30 62. ( ) − ° sin 135 63. ( ) − ° sec 60 64. ( ) − ° csc 30 65. ( ) − ° sin 90 66. ( ) − ° cos 270 67. π ( ) − tan 4 68. π ( ) − sin 69. π ( ) − cos 4 70. π ( ) − sin 3 71. π ( ) − tan 72. π ( ) − sin 3 2 73. π ( ) − csc 4 74. π ( ) − sec 75. π ( ) − sec 6 76. π ( ) − csc 3 In Problems 77–88, use properties of the trigonometric functions to find the exact value of each expression. Do not use a calculator. 77. ° + ° sin 40 cos 40 2 2 78. ° − ° sec 18 tan 18 2 2 79. ° sin80 ° csc80 80. ° tan10 ° cot10 81. ° − ° ° tan40 sin40 cos40 82. ° − ° ° cot20 cos20 sin20 83. ° ⋅ ° cos400 sec40 84. ° ⋅ ° tan200 cot20 85. π π ( ) − sin 12 csc 25 12 86. π π ( ) − ⋅ sec 18 cos 37 18 87. ( ) − ° ° + ° sin 20 cos380 tan200 88. ( ) ( ) ° − ° + − ° sin70 cos 430 tan 70 89. If θ = sin 0.3, find the value of: θ θ π θ π ( ) ( ) + + + + sin sin 2 sin 4 90. If θ = cos 0.2, find the value of: θ θ π θ π ( ) ( ) + + + + cos cos 2 cos 4 91. If θ = tan 3, find the value of: θ θ π θ π ( ) ( ) + + + + tan tan tan 2 92. If θ = − cot 2, find the value of: θ θ π θ π ( ) ( ) + − + − cot cot cot 2 93. Find the exact value of: °+ °+ °++ °+ ° sin1 sin2 sin3 sin358 sin359 94. Find the exact value of: cos1 cos2 cos3 cos358 cos359 °+ °+ °++ °+ ° 95. What is the domain of the sine function? 96. What is the domain of the cosine function? 97. For what numbers θ is θ θ ( ) = f tan not defined? 98. For what numbers θ is θ θ ( ) = f cot not defined? 99. For what numbers θ is θ θ ( ) = f sec not defined? 100. For what numbers θ is θ θ ( ) = f csc not defined? 101. What is the range of the sine function? 102. What is the range of the cosine function? 103. What is the range of the tangent function?
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