Question

The number of solution of the equation $\sin x+\sin 5x=\sin 3x$ tying in the interval $[0,\pi ]$ is

• A. $4$
• B. $6$
• C. $5$
• D. $2$

Answer 1

Answer: (B)

$\sin x+\sin 5x=\sin 3x$
$\Rightarrow$ $2\sin \left(\frac{x+5x}{2}\right).\cos \left(\frac{5x-x}{2}\right)=\sin 3x$
$\Rightarrow$ $2\sin 3x.cos2x=sin3x$
$\Rightarrow$ $\sin 3x(2\cos 2x-1)=0$
$\Rightarrow$ $\sin 3x=0$ and $2\cos 2x-1=0$
$\Rightarrow$ $\sin 3x=sin0$ and $\cos 2x=\frac{1}{2}=\cos \frac{\pi }{3}$
$\Rightarrow$ $3x=n\pi$ and $2x=2n\pi \pm \frac{\pi }{3}$
$\Rightarrow$ $x=\frac{n\pi }{3}$ and $x=n\pi \pm \frac{\pi }{6}$
where $n\in z,$
$\Rightarrow$ $x=0,\frac{\pi }{3},\frac{2\pi }{3},\frac{3\pi }{3}$
and $x=\frac{\pi }{6},\frac{5\pi }{6}$
For $n=0,1,2,\mathrm{3....}$
So, the total number of solution lie between
$[0,\pi ]$ $x=0,\frac{\pi }{6},\frac{\pi }{3},\frac{2\pi }{3},\frac{5\pi }{6},\frac{3\pi }{3}$