Use the given information to find the exact value of each of the following. a. s

Question

Use the given information to find the exact value of each of the following. a. sin 2 theta b. cos 2 theta c. tan 2 theta cot theta = 12, theta lies in quadrant III a. sin 2 theta = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) b. cos 2 theta = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) c. tan 2 theta = (Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

Explanation / Answer

cot theta = 12

base / perpendicular = 12/1

hypotenuse = sqrt (12^2 + 1) = sqrt 145

sin theta = -1 / sqrt 145

cos theta = -12 / sqrt 145

sin ( 2 theta ) = 2 sin theta cos theta

= 2 ( -1/ sqrt 145 ) ( -12 / sqrt 145 )

= 24 / 145

cos ( 2 theta ) = cos^2 theta - sin^2 theta

= 144 / 145 - 1 / 145

= 143 / 145

tan ( 2 theta ) = 2 tan theta / ( 1 - tan^2 theta )

= 2 ( 1/12) / 1 - ( 1/ 144 )

1/6 / 143 / 144

= 24 / 143

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