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  4. The possible values of θ ∈ (0, π) such that sin(θ ) + sin(4 θ ) + sin(7 θ ) = 0 are :

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Q. The possible values of $\theta \in \left(0, \pi\right)$ such that sin($\theta $) + $sin(4\,\theta $) + $sin(7\,\theta ) = 0$ are :

AIEEEAIEEE 2011Trigonometric Functions

Solution:

$sin 4\theta+ 2sin\, 4 \,\theta \,cos \,3\theta = 0\quad\because\quad\theta,\,\in\left(0, \pi\right)$
$sin\, 4\theta\left(1 + 2 \,cos \,3\, \theta\right) = 0$
$sin \,4\theta= 0 \quad$ or$\quad cos \,3\theta= - \frac{1}{2}$
$4\theta = n\pi ; n\in I \quad$ or$\quad3\theta = 2n\pi \pm \frac{2\pi}{3}, n \in I$
$\theta = \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4} \quad$ or $\quad\theta = \frac{2\pi}{9}, \frac{8\pi}{9}, \frac{4\pi}{9}$