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Trigonometrical Equations
medium
$\sin 6\theta + \sin 4\theta + \sin 2\theta = 0,$ then $\theta = $
A
$\frac{{n\pi }}{4}$ 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
Solution
(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 $n\pi \pm \frac{\pi }{3}$.
Standard 11
Mathematics
