Question

The number of values of $x$ between 0 and $2\pi$ such that the equation $\sin x+\sin 2x+\sin 3x=\cos x+\cos 2x+\cos 3x$ must be ___

Answer 1

$(\sin 3x+\sin x)+\sin 2x=(\cos 3x+\cos x)+\cos 2x$
$\Rightarrow 2\sin 2x\cos x+\sin 2x=2\cos 2x\cos x+\cos 2x$
$\Rightarrow \sin 2x(2\cos x+1)-\cos 2x(2\cos x+1)=0$
$\Rightarrow (2\cos x+1)(\sin 2x-\cos 2x)=0$
$\Rightarrow \cos x=-\frac{1}{2},\tan 2x=1$
$\therefore x=\frac{2\pi }{3},\frac{4\pi }{3}$ or $\frac{\pi }{8},\frac{5\pi }{8},\frac{9\pi }{8},\frac{13\pi }{8}$