Question

The number of solution(s) of $|\cos 3\theta |=1$, where $\theta \in [-\pi ,\pi ]$ is

Answer 1

Given: $|\cos 3\theta |=1$
As $\theta \in [-\pi ,\pi ]\Rightarrow 3\theta \in [-3\pi ,3\pi ]$
$\Rightarrow \cos 3\theta =\pm 1$

When $\cos 3\theta =1$
$\Rightarrow 3\theta =-2\pi ,0,2\pi (\because 3\theta \in [-3\pi ,3\pi \left]\right)\Rightarrow \theta =-\frac{2\pi }{3},0,\frac{2\pi }{3}$

When $\cos 3\theta =-1$
$\Rightarrow 3\theta =-3\pi ,-\pi ,\pi ,3\pi \Rightarrow \theta =-\pi ,-\frac{\pi }{3},\frac{\pi }{3},\pi$
Therefore,
$\theta =-\pi ,-\frac{2\pi }{3},-\frac{\pi }{3},0,\frac{\pi }{3},\frac{2\pi }{3},\pi$

Hence, total number of solutions is $7$