SECTION 6.2 Trigonometric Functions: Unit Circle Approach 409 In Problems 65–76, use a calculator to find the approximate value of each expression rounded to two decimal places. 65. sin28° 66. cos14° 67. sec21° 68. cot70° 69. tan 10 π 70. sin 8 π 71. cot 12 π 72. csc 5 13 π 73. sin1 74. tan 1 75. sin1° 76. tan1° In Problems 77–84, a point on the terminal side of an angle θ in standard position is given. Find the exact value of each of the six trigonometric functions of .θ 77. 3, 4 ( ) − 78. 5, 12 ( ) − 79. 2, 3 ( ) − 80. 1, 2 ( ) − − 81. 2, 2 ( ) − − 82. 1, 1 ( ) − 83. 1 3 , 1 4 ( ) 84. 0.3, 0.4 ( ) 85. Find the exact value of: sin45 sin135 sin225 sin315 ° + ° + ° + ° 86. Find the exact value of: tan60 tan150 ° + ° 87. Find the exact value of: sin40 sin130 sin220 sin310 ° + ° + ° + ° 88. Find the exact value of: tan40 tan140 ° + ° 89. If f sin 0.1, θ θ ( ) = = find f . θ π ( ) + 90. If f cos 0.3, θ θ ( ) = = find f . θ π ( ) + 91. If f tan 3, θ θ ( ) = = find f . θ π ( ) + 92. If f cot 2, θ θ ( ) = = − find f . θ π ( ) + 93. If sin 1 5 , θ = find csc .θ 94. If cos 2 3 , θ = find sec .θ In Problems 95–106, f sin θ θ ( ) = and g cos . θ θ ( ) = Find the exact value of each function below if 60 . θ = ° Do not use a calculator. 95. f θ( ) 96. g θ( ) 97. f 2 θ( ) 98. g 2 θ( ) 99. f 2 θ [ ] ( ) 100. g 2 θ [ ] ( ) 101. f 2θ ( ) 102. g 2θ ( ) 103. f2 θ( ) 104. g2 θ( ) 105. f θ ( ) − 106. g θ ( ) − Mixed Practice In Problems 107–116, f x x g x x h x x p x x sin , cos , 2 , and 2 . ( ) ( ) ( ) ( ) = = = = Find the value of each of the following: 107. f g 30 ( )( ) + ° 108. f g 60 ( )( ) − ° 109. f g 3 4 π ( ) ( ) ⋅ 110. f g 4 3 π ( ) ( ) ⋅ 111. f h 6 π( ) ( ) 112. g p 60 ( )( )° 113. p g 315 ( )( )° 114. h f 5 6 π ( ) ( ) 115. (a) Find f 4 . π( ) What point is on the graph of f? (b) Assuming x f 0 2 , π ≤ ≤ is one-to-one.* Use the result of part (a) to find a point on the graph of f .1− (c) What point is on the graph of y f x x 4 3 if 4 ? π π ( ) = + − = 116. (a) Find g 6 . π( ) What point is on the graph of g? (b) Assuming x g 0 2 , π ≤ ≤ is one-to-one.* Use the result of part (a) to find a point on the graph of g .1− (c) What point is on the graph of y g x x 2 6 if 6 ? π π ( ) = − = 119. Use a calculator in radian mode to complete the following table. What do you conjecture about the value of f sin θ θ θ ( ) = as θ approaches 0? θ 0.5 0.4 0.2 0.1 0.01 0.001 0.0001 0.00001 sinθ f sin θ θ θ ( ) = Applications and Extensions 117. Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is 1 2 , 3 2 . ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ 118. Find two negative and three positive angles, expressed in radians, for which the point on the unit circle that corresponds to each angle is 2 2 , 2 2 . − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ *In Section 7.1, we discuss the necessary domain restriction so that the function is one-to-one.
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