Question

Find all the values of $\theta$ which satisfy the equation: $\cos \theta .\cos 2\theta .\cos 3\theta =1/4$.

Answer 1

$cos\theta cos2\theta cos3\theta =\frac{1}{4}$

$\Rightarrow \frac{1}{2}\left(2cos3\theta cos\theta \right)cos2\theta =\frac{1}{4}$

$\Rightarrow (cos4\theta +cos2\theta )cos2\theta =\frac{1}{2}$

$\Rightarrow cos4\theta cos2\theta +co{s}^{2}2\theta =\frac{1}{2}$

$\Rightarrow cos4\theta cos2\theta +\frac{1+cos4\theta }{2}=\frac{1}{2}$

$\Rightarrow cos4\theta cos2\theta +\frac{1}{2}cos4\theta =0$

$\Rightarrow cos4\theta (cos2\theta +\frac{1}{2})=0$

$cos4\theta =0$, $cos2\theta =\frac{-1}{2}$

$\Rightarrow 4\theta =(2n+1)\frac{\pi }{2}$, $cos2\theta =cos\frac{2\pi }{3}$

$\theta =(2n+1)\frac{\pi }{8},$ $2\theta =2n\pi \pm \frac{2\pi }{3}$

$\theta =(2n+1)\frac{\pi }{8},$ $\theta =n\pi \pm \frac{\pi }{3}$