Question

Solve $\cos \theta +cos2\theta +cos3\theta =0$

Answer 1

Given, $\cos \theta +cos2\theta +cos3\theta =0\Rightarrow \cos \theta +\cos 3\theta +\cos 2\theta =0$
$\Rightarrow 2\cos \left(\frac{\theta +3\theta }{2}\right)\cos \left(\frac{\theta -3\theta }{2}\right)+\cos 2\theta =0$
$\Rightarrow 2\cos 2\theta \cos \theta +\cos 2\theta =0\Rightarrow \cos 2\theta (2\cos \theta +1)=0$
$\Rightarrow \cos 2\theta =0$ or $\cos \theta =-\frac{1}{2}$
$\Rightarrow 2\theta =\frac{(2n+1)\pi }{2}$ or $\theta =2n\pi \pm \frac{2\pi }{3}$
$\Rightarrow \theta =\frac{(2n+1)\pi }{4}$ or $\theta =2n\pi \pm \frac{2\pi }{3}$