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85 sin 75 cos x 86 3 cos sin 10 x π 87 2 sec csc 5 x

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85. ( ) ( ) sin 75 cos x = ! 86. ( ) 3 cos sin 10 x π = 87. ( ) 2 sec csc 5 x π = 88. ( ) ( ) csc 41 sec x = ! 89. ( ) ( ) tan 72 cot x = ! 90. ( ) cot tan 3 x π = Simplify the following. 91. ( ) ( ) cos 90 csc x x ! 92. ( ) sin sec 2 x x π 93. ( ) sin tan 2 x x π 94. ( ) csc sin 2 x x π 95. ( ) ( ) sec 90 cos θ θ ! 96. ( ) ( ) tan 90 csc 90 x x ! ! 97. ( ) ( ) sec 20 sin 70 ! ! 98. cos csc 6 3 π π 99. 2 2 5 cot sec 12 12 π π 100. ( ) ( ) 2 2 cos 72 cos 18 + ! !
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Exercise Set 6.2: Double-Angle and Half-Angle Formulas University of Houston Department of Mathematics 94 Answer the following. 1. (a) Evaluate sin 2 6 π . (b) Evaluate 2sin 6 π . (c) Is sin 2 2sin 6 6 π π = ? (d) Graph ( ) ( ) sin 2 f x x = and ( ) ( ) 2sin g x x = on the same set of axes. (e) Is ( ) ( ) sin 2 2sin x x = ? 2. (a) Evaluate cos 2 6 π . (b) Evaluate 2cos 6 π . (c) Is cos 2 2cos 6 6 π π = ? (d) Graph ( ) ( ) cos 2 f x x = and ( ) ( ) 2cos g x x = on the same set of axes. (e) Is ( ) ( ) cos 2cos x x = ? 3. (a) Evaluate tan 2 6 π . (b) Evaluate 2 tan 6 π (c) Is tan 2 2tan 6 6 π π = ? (d) Graph ( ) ( ) tan 2 f x x = and ( ) ( ) 2tan g x x = on the same set of axes. (e) Is ( ) ( ) tan 2 2 tan x x = ? 4. Derive the formula for ( ) sin 2 θ by using a sum formula on ( ) sin θ θ + . 5. Derive the formula for ( ) cos 2 θ by using a sum formula on ( ) cos θ θ + . 6. Derive the formula for ( ) tan 2 θ by using a sum formula on ( ) tan θ θ + . 7. The sum formula for cosine yields the equation ( ) ( ) ( ) 2 2 cos 2 cos sin θ θ θ = . To write ( ) cos 2 θ strictly in terms of the sine function, (a) Using the Pythagorean identity ( ) ( ) 2 2 cos sin 1 θ θ + = , solve for ( ) 2 cos θ . (b) Substitute the result from part (a) into the above equation for ( ) cos 2 θ . 8. The sum formula for cosine yields the equation ( ) ( ) ( ) 2 2 cos 2 cos sin θ θ θ = . To write ( ) cos 2 θ strictly in terms of the cosine function, (a) Using the Pythagorean identity ( ) ( ) 2 2 cos sin 1 θ θ + = , solve for ( ) 2 sin θ . (b) Substitute the result from part (a) into the above equation for ( ) cos 2 θ . Answer the following. 9. Suppose that ( ) 12 cos 13 α = and 3 2 2 π α π < < . Find: (a) ( ) sin 2 α (b) ( ) cos 2 α (c) ( ) tan 2 α 10. Suppose that ( ) 3 tan 4 α = and 3 2 π π α < < . Find: (a) ( ) sin 2 α (b) ( ) cos 2 α (c) ( ) tan 2 α
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