532 CHAPTER 7 Analytic Trigonometry 47. Find an expression for θ ( ) cos 3 as a third-degree polynomial in the variable θ cos . 48. Find an expression for θ ( ) cos 4 as a fourth-degree polynomial in the variable θ cos . 49. Find an expression for θ ( ) sin 5 as a fifth-degree polynomial in the variable θ sin . 50. Find an expression for θ ( ) cos 5 as a fifth-degree polynomial in the variable θ cos . In Problems 51–72, establish each identity. 51. θ θ θ ( ) − = cos sin cos 2 4 4 52. θ θ θ θ θ ( ) − + = cot tan cot tan cos 2 53. θ θ θ ( ) = − cot 2 cot 1 2 cot 2 54. θ θ θ ( ) ( ) = − cot 2 1 2 cot tan 55. θ θ θ ( ) = − sec 2 sec 2 sec 2 2 56. θ θ θ ( ) = csc 2 1 2 sec csc 57. ( ) ( ) ( ) − = u u u cos 2 sin 2 cos 4 2 2 58. ( ) ( ) ( ) − = u u u u 4 sin cos 1 2 sin sin 4 2 59. θ θ θ θ ( ) ( ) + = − + cos 2 1 sin 2 cot 1 cot 1 60. θ θ θ ( ) = sin cos 1 4 sin 2 2 2 2 61. θ θ = + sec 2 2 1 cos 2 62. θ θ = − csc 2 2 1 cos 2 63. = + − v v v cot 2 sec 1 sec 1 2 64. = − v v v tan 2 csc cot 65. θ θ θ = − + cos 1 tan 2 1 tan 2 2 2 66. θ θ θ θ θ ( ) − = + + 1 1 2 sin 2 sin cos sin cos 3 3 67. θ θ θ θ ( ) ( ) − = sin 3 sin cos 3 cos 2 68. θ θ θ θ θ θ θ θ θ ( ) + − − − + = cos sin cos sin cos sin cos sin 2 tan 2 69. θ θ θ θ ( ) = − − tan 3 3 tan tan 1 3 tan 3 2 70. θ θ θ θ ( ) ( ) ( ) + + ° + + ° = tan tan 120 tan 240 3 tan 3 71. θ θ ( ) ( ) = − − ln sin 1 2 ln 1 cos 2 ln 2 72. θ θ ( ) ( ) = + − ln cos 1 2 ln 1 cos 2 ln 2 In Problems 73–82, solve each equation on the interval θ π ≤ < 0 2 . 7 3. θ θ ( ) + = cos 2 6 sin 4 2 74. θ θ ( ) = − cos 2 2 2 sin2 75. θ θ ( ) = cos 2 cos 76. θ θ ( ) = sin 2 cos 77. θ θ ( ) ( ) + = sin 2 sin 4 0 78. θ θ ( ) ( ) + = cos 2 cos 4 0 79. θ θ ( ) − = 3 sin cos 2 80. θ θ ( ) + + = cos 2 5 cos 3 0 81. θ θ ( ) + = tan 2 2 sin 0 82. θ θ ( ) + = tan 2 2 cos 0 In Problems 83–94, find the exact value of each expression. 83. ( ) − sin 2 sin 1 2 1 84. ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ − sin 2 sin 3 2 1 85. ( ) − cos 2 sin 3 5 1 86. ( ) − cos 2 cos 4 5 1 87. ( ) − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − tan 2 cos 3 5 1 88. ( ) − tan 2 tan 3 4 1 89. ( ) − sin 2 cos 4 5 1 90. ( ) − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − cos 2 tan 4 3 1 91. ( ) − sin 1 2 cos 3 5 2 1 92. ( ) − cos 1 2 sin 3 5 2 1 93. ( ) − sec 2 tan 3 4 1 94. ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − csc 2 sin 3 5 1 Applications and Extensions In Problems 95–100, find the real zeros of each trigonometric function on the interval θ π ≤ < 0 2 . 95. ( ) ( ) = − f x x x sin 2 sin 96. ( ) ( ) = + f x x x cos 2 cos 97. ( ) ( ) = + f x x x cos 2 sin2 98. ( ) ( ) = − f x x x 2 sin sin 2 2 99. ( ) ( ) = + f x x x sin 2 cos 100. ( ) ( ) = − − f x x x cos 2 5 cos 2 101. Area of an Octagon (a) The area A of a regular octagon is given by the formula π = A r8 tan 8 , 2 where r is the apothem, which is a line segment from the center of the octagon perpendicular to a side. See the figure. Find the exact area of a regular octagon whose apothem is 12 inches. a r
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