MCQ
$\sin 6\theta + \sin 4\theta + \sin 2\theta = 0,$ then $\theta = $
  • $\frac{{n\pi }}{4}$ or $n\pi \pm \frac{\pi }{3}$
  • B
    $\frac{{n\pi }}{4}$ or $n\pi \pm \frac{\pi }{6}$
  • C
    $\frac{{n\pi }}{4}$ or $2n\pi \pm \frac{\pi }{6}$
  • D
    None of these

Answer

Correct option: A.
$\frac{{n\pi }}{4}$ or $n\pi \pm \frac{\pi }{3}$
a
(a) $\sin 6\theta + \sin 4\theta + \sin 2\theta = 0$

$ \Rightarrow 2\sin 4\theta \cos 2\theta + \sin 4\theta = 0$

$ \Rightarrow $$\sin 4\theta (2\cos 2\theta + 1) = 0$

$ \Rightarrow $ $2\cos 2\theta = - 1$ 

$ \Rightarrow $ $\cos 2\theta = - \frac{1}{2}$

$ \Rightarrow $ $2\theta = 2n\pi \pm \frac{{2\pi }}{3} $

$\Rightarrow \theta = n\pi \pm \frac{\pi }{3}$

and $\sin 4\theta = 0 \Rightarrow 4\theta = n\pi \Rightarrow \theta = \frac{{n\pi }}{4}$

$\theta = \frac{{n\pi }}{4}$ or $n\pi \pm \frac{\pi }{3}$.

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