Question

Number of solutions of equation $\sin 9\theta =\sin \theta$ in the interval $[0,2\pi ]$ is

• A. 16
• B. 17
• C. 18
• D. 15

Answer 1

Answer: (B)

$\sin 9\theta =\sin \theta$
$\Rightarrow 9\theta =n\pi +(-1{)}^n\theta$
If $n=2m$ then $9\theta =2m\pi +\theta$
$\Rightarrow \theta =\frac{m\pi }{4}$
If $n=2m+1$ then $9\theta =(2m+1)\pi =-\theta$
$\Rightarrow \theta =(2m+1)\frac{\pi }{10}$
The values belonging to $[0,\pi ]$ are
$\theta =0,\frac{\pi }{10},\frac{\pi }{4},\frac{3\pi }{10},\frac{\pi }{2},\frac{7\pi }{10},\frac{3\pi }{4},\frac{9\pi }{10}$, $\pi ,\frac{11\pi }{10},\frac{5\pi }{4},\frac{13\pi }{10},\frac{3\pi }{2},\frac{17\pi }{10},\frac{7\pi }{4},\frac{19\pi }{10},2\pi$