Question

The number of solutions of the equation $\sin\, (9x) + \sin\, (3x) - 0$ in the closed interval $[0,2 \pi]$ is

• A. 7
• B. 13
• C. 19
• D. 25

Answer 1

Answer: (B)

Given trigonometric equation is
$\sin (9 x)+\sin (3 x)=0$
$\Rightarrow 2 \sin 6 x \cos 3 x=0$
$\therefore $ either $\sin 6 x=0$
or $\cos 3 x=0$
for $x \in[0,2 \pi]$
$\sin 6 x=0$
$\Rightarrow x=0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{5 \pi}{6}$,
$\pi, \frac{7 \pi}{6}, \frac{4 \pi}{3}, \frac{3 \pi}{2}, \frac{5 \pi}{3}, 2 \pi$
and $3 x=0$
$\Rightarrow x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{3 \pi}{2}, \frac{11 \pi}{16}$