Question

If $\sin 6\theta + \sin 4\theta + \sin 2\theta = 0$, then general value of $\theta$ is

• A. $\frac{n\pi}{4},n\pi\pm\frac{\pi}{3}$
• B. $\frac{n\pi}{4},n\pi\pm\frac{\pi}{4}$
• C. $\frac{n\pi}{4},n\pi\pm\frac{\pi}{6}$
• D. $\frac{n\pi}{4},2n\pi\pm\frac{\pi}{6}$

Answer 1

Answer: (A)

$\sin 6\theta + \sin 4\theta + \sin 2\theta = 0$
$\Rightarrow \, \, \, \, \, \, (\sin 6\theta + \sin \: 2\theta) + \sin 4\theta = 0$
$\Rightarrow \, \, \, \, \, 2 \sin 4\theta \cos 2\theta + \sin 4\theta = 0$
$\Rightarrow \, \, \, \, \, \, \sin 4\theta (2 \cos 2\theta + 1) = 0$
$\Rightarrow \, \, \, \, \, \, \, \, \sin 4\theta = 0$ or $\cos 2\theta = -\frac{1}{2} = \cos \frac{2 \pi}{3} $
$\Rightarrow \, \, \, \, \, \, 4\theta = n \pi $ or $2 \theta = 2 n \pi \pm \frac{2}{3}$
$\Rightarrow \, \, \, \, \, \, \theta = \frac{n \pi}{4} , \theta = n \pi \pm \frac{\pi}{3} \, \theta \, $