Question

Number of solutions of the equation $|\cos 3\theta |=1$ in $[-\pi ,\pi ]$ is

Answer 1

$|\cos 3\theta |=1$
$\Rightarrow \cos 3\theta =\pm 1$

When $\cos 3\theta =1$
$\Rightarrow 3\theta =2n\pi ,n\in Z$
$\Rightarrow \theta =\frac{n\pi }{3},n\in Z$
Number of solutions in $[-\pi ,\pi ]$ are $3.$
i.e. $\{-\frac{2\pi }{3},0,\frac{2\pi }{3}\}$

When $\cos 3\theta =-1$
$\Rightarrow 3\theta =(2n+1)\pi ,n\in Z$
$\Rightarrow \theta =\left(\frac{2n+1}{3}\right)\pi ,n\in Z$
Number of solutions in $[-\pi ,\pi ]$ are $4.$
i.e. $\{-\pi ,-\frac{\pi }{3},\frac{\pi }{3},\pi \}$

Hence, total number of solutions $=7$

Alternate Solution:
$|\cos \theta |$ is periodic function with period of $\pi$
Hence period of $|\cos 3\theta |$ is $\frac{\pi }{3}$

Number of solution in $(0,\pi /3]$is $1$
So number of solution in $(0,\pi ]$ is $3$
similarly no of solution $[-\pi ,0)$ is $3$
here, $\theta =0$ is also a solution
$\therefore$ total number of solution is $7$