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Trigonometrical Equations
medium
The values of $\theta $ satisfying $\sin 7\theta = \sin 4\theta - \sin \theta $ and $0 < \theta < \frac{\pi }{2}$ are
A
$\frac{\pi }{9},\frac{\pi }{4}$
B
$\frac{\pi }{3},\frac{\pi }{9}$
C
$\frac{\pi }{6},\frac{\pi }{9}$
D
$\frac{\pi }{3},\frac{\pi }{4}$
Solution
(a) $\sin 7\theta + \sin \theta – \sin 4\theta = 0$
$ \Rightarrow $ $2\sin 4\theta \cos 3\theta – \sin 4\theta = 0$
$ \Rightarrow $ $\sin 4\theta (2\cos 3\theta – 1) = 0 $
$\Rightarrow \sin 4\theta = 0,\,{\rm{ }}\cos 3\theta = \frac{1}{2}$
Now $\sin 4\theta = 0$ $ \Rightarrow $ $4\theta = \pi $ $ \Rightarrow $ $\theta = \frac{\pi }{4}$.
and $\cos 3\theta = \frac{1}{2}$ $ \Rightarrow $ $3\theta = \frac{\pi }{3}$ $ \Rightarrow $ $\theta = \frac{\pi }{9}$.
Standard 11
Mathematics
