Question

The number of solutions of $\sin \, x + \sin \, 3x + \sin \, 5x = 0$ in the interval $\left[\frac{\pi}{2} , 3 \frac{\pi}{2}\right] $ is

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The Correct Option is B

We have,
$ \sin x+\sin 3 x+\sin 5 x=0 $
$\Rightarrow \sin x+\sin 5 x+\sin 3 x=0$
$\Rightarrow 2 \sin 3 x \cos 2 x+\sin 3 x=0 $
$\Rightarrow \sin 3 x(2 \cos 2 x+1)=0 $
$\Rightarrow \sin 3 x=0 $ or $ \cos 2 x=-\frac{1}{2}=\cos \frac{2 \pi}{3} $
$\Rightarrow 3 x=n \pi \text { or } 2 n=2 n \pi \pm \frac{2 \pi}{3} $
$\Rightarrow x=\frac{n \pi}{3} $ or $ x=n \pi \pm \frac{\pi}{3}$
But, it is given that
$x \in\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
$\therefore x=\frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}$
$\therefore$ Number of solutions is 3 .