Use the given information to find the exact value of each of

Use the given information to find the exact value of each of the following, sin 2theta b cos 2theta tan 2theta cot theta = 12. theta lies in quadrant III sin 2theta = (Simplify your answer. Type an exact answer, using radicals as needed Use integers or fractions for any numbers in the expression. Rationalize all denominators.) cos2theta = (Simplify your answer. Type an exact answer, using radicals as needed Use integers or fractions for any numbers in the expression. Rationalize all denominators.) tan 2theta = (Simplify your answer. Type an exact answer, using radicals as needed Use integers or fractions for any numbers in the expression. Rationalize all denominators.)

Solution

since theta is in 3rd quadrant cot(theta) must be negative but it is given positive in question.

assuming cot(theta) = 12,

tan(theta) = 1/12

sec(theta) = sqrt(1+tan^2(theta)) = sqrt(145)/12

cos(theta) = 12/sqrt(145)

sin(theta) = -sqrt(1-cos^2(theta)) = -1/12

a)sin2(theta) = 2sin(theta)cos(theta) = -0.167244

b)cos2(theta) = 1-2sin^2(theta) = 1-2/144 = 0.986111

c)tan2(theta) = sin2(theta)/cos2(theta)=-0.169599