Question

Find the general solution:
$sinx+sin3x+sin5x=0$

Answer 1

$sinx+sin3x+sin5x=0$

$\Rightarrow sinx+sin5x+sin3x=0$

$\Rightarrow 2\sin \left(\frac{x+5x}{2}\right)\cos \left(\frac{x-5x}{2}\right)+sin3x=0\Rightarrow \sin 3x(2\cos 2x+1)=0$

$\Rightarrow \sin 3x=0$ or $\cos 2x=\frac{-1}{2}$

$\Rightarrow 3x=n\pi$ or $2x=2n\pi \pm \frac{\pi }{3}$

$\Rightarrow x=\frac{n\pi }{3}$, $n\pi \pm \frac{\pi }{6}$