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