Question

Number of solution(s) of $\sin 5x+\sin 3x+\sin x=0$ in $0\le x\le \pi$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

$\sin 3x+(\sin 5x+\sin x)=0$
$\Rightarrow \sin 3x+\left(2\sin 3x\cos 2x\right)=0$
$\Rightarrow \sin 3x(2\cos 2x+1)=0$
$\Rightarrow \sin 3x=0\text{or}\cos 2x=-\frac{1}{2}=\cos \frac{2\pi }{3}$
$\Rightarrow x=\frac{n\pi }{3},x=n\pi \pm \frac{\pi }{3},n\in Z$
$\therefore x=0,\frac{\pi }{3},\frac{2\pi }{3},\pi$ are the four solutions in $0\le x\le \pi$.