Question

Given tanθ=3/2 and 180°<θ<270° , find the exact value of each expression.

a. cos 2θ

Answer 1

`180^(\circ) < \theta < 270^(\circ) \implies \cos \theta, sec \theta < 0 \\ \\ \\ 1+\tan^2 \theta=\sec^2 \theta \\ \\ 1+(3/2)^2=\sec^2 \theta \\ \\ \sec \theta=-(√(13))/(2) \\ \\ \implies \cos \theta=-(2)/(\sqrt 13) \\ \\ \\ \cos 2\theta=2\cos^2 \theta-1=2 \left(-(2)/(\sqrt 13) \right)^2-1=\boxed{-(5)/(13)}`