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)

17

Answer 2

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