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  4. The values of θ satisfying sin 7θ = sin 4θ - sin θ and 0 < θ <(π/2) are

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Q. The values of $ \theta$ satisfying $\sin 7\theta = \sin 4\theta - \sin \theta$ and $0 < \theta <\frac{\pi}{2}$ are

Trigonometric Functions

Solution:

$\sin\,7\,\theta=\sin\,4\,\theta-\sin\,\theta$
$\Rightarrow \,\sin\,7\,\theta+\sin\,\theta +\sin\,\theta=\sin\,4\,\theta$
$\Rightarrow \,2\,\sin\,4\,\theta\,\cos\,3\,\theta=\sin\,4\,\theta$
$\Rightarrow \,\sin\,4\theta(2\,\cos\,3\theta-1)=0$
$ \Rightarrow \,\sin\,4\,\theta=0$ or $\cos\,3\theta=\frac{1}{2}$
$4\theta=\pi,2\pi,3\pi.....or\,3\theta=\frac{\pi}{3},\frac{5\pi}{3}.....$
$\Rightarrow \,\theta=\frac{\pi}{4},$ and $\theta\,\frac{\pi}{9}\,\in\,\left(0,\frac{\pi}{2}\right)$
Thus $\theta=\frac{\pi}{4},\frac{\pi}{9}$