Question

Solve $\sin \theta +\sin 2\theta +\sin 3\theta +\sin 4\theta =0$.

Answer 1

$(\sin \theta +\sin 3\theta )+(\sin 2\theta +\sin 4\theta )=0$

$2\sin 2\theta \cos \theta +2\sin 3\theta \cos \theta =0$

$2\cos \theta (\sin 2\theta +\sin 3\theta )=0$

$4\cos \theta \sin \left(5\theta /2\right)\cos \left(\theta /2\right)=0$

$\cos \theta =0=\cos \left(\pi /2\right)$,

$\therefore \theta =(n+\frac{1}{2})\pi$

$\cos \left(\theta /2\right)=0=\cos \left(\pi /2\right)$,

$\therefore \frac{\theta }{2}=(n+\frac{1}{2})\pi$ or $\theta =(2n+1)\pi$

$\sin \left(5\theta /2\right)=0=\sin 0$

$\therefore \left(5\theta /2\right)=n\pi$ or $\theta =2n\pi /5$.