solving equations?

Question

Solve the following equations on the interval 0 < (equal to) theta < 2 pi


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


b) cos (2 theta) + 5 cos theta + 3 = 0


please help and explain this to me step by step I keep getting weird answers, thanks

Answer 1

They're both trick questions.

Cos 2theta = 1 - 2 sin^2 theta. So a doesn't have any solutions.

cos 2 theta also = 2 cos^2 theta - 1. So b is a simple quadratic equation in cos theta, namely 2c^2 + 5c +2. But that's 2 (c+1)^2 + 1, which always equals at least 1 since the square of anything is at least 0. So b doesn't have any solutions either.