The values of theta satisfying sin 7theta = s | vedclass

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Question

(A) Given the equation: $\sin 7	heta = \sin 4	heta - \sin 	heta$Rearranging the terms: $\sin 7	heta + \sin 	heta = \sin 4	heta$Using the sum-to-

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$\frac{\pi}{9}, \frac{\pi}{4}$

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(A) Given the equation: $\sin 7	heta = \sin 4	heta - \sin 	heta$Rearranging the terms: $\sin 7	heta + \sin 	heta = \sin 4	heta$Using the sum-to-product formula $\sin C + \sin D = 2 \sin(\frac{C+D}{2}) \cos(\frac{C-D}{2})$:$2 \sin(\frac{7	heta + 	heta}{2}) \cos(\frac{7	heta - 	heta}{2}) = \sin 4	heta$$2 \sin 4	heta \cos 3	heta = \sin 4	heta$$2 \sin 4	heta \cos 3	heta - \sin 4	heta = 0$$\sin 4	heta (2 \cos 3	heta - 1) = 0$This gives two cases:Case $1$: $\sin 4	heta = 0$$4	heta = n\pi$,so $	heta = \frac{n\pi}{4}$. For $0 Case $2$: $2 \cos 3	heta - 1 = 0 \Rightarrow \cos 3	heta = \frac{1}{2}$$3	heta = 2n\pi \pm \frac{\pi}{3}$. For $0 Thus,the values are $\frac{\pi}{9}$ and $\frac{\pi}{4}$.

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

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