If sin theta + sin 5 theta = sin 3 theta and | vedclass

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(A) Given equation: $\sin 	heta + \sin 5 	heta = \sin 3 	heta$Using the sum-to-product formula $\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A

Vedclass · Accepted Answer

$30$

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(A) Given equation: $\sin 	heta + \sin 5 	heta = \sin 3 	heta$Using the sum-to-product formula $\sin A + \sin B = 2 \sin \frac{A+B}{2} \cos \frac{A-B}{2}$:$2 \sin \frac{	heta + 5 	heta}{2} \cos \frac{5 	heta - 	heta}{2} = \sin 3 	heta$$2 \sin 3 	heta \cos 2 	heta = \sin 3 	heta$Rearranging the terms:$2 \sin 3 	heta \cos 2 	heta - \sin 3 	heta = 0$$\sin 3 	heta (2 \cos 2 	heta - 1) = 0$This gives two cases:Case $1$: $\sin 3 	heta = 0$. Since $0 Case $2$: $2 \cos 2 	heta - 1 = 0 \implies \cos 2 	heta = \frac{1}{2} = \cos 60^{\circ}$.$2 	heta = 60^{\circ} \implies 	heta = 30^{\circ}$.Checking the range $0 < 	heta < 90^{\circ}$,both $30^{\circ}$ and $60^{\circ}$ are valid. However,usually,the smallest positive value is sought. Given the options,$30^{\circ}$ is the correct answer.

If $\sin \theta + \sin 5 \theta = \sin 3 \theta$ and $0 < \theta < \frac{\pi}{2}$,what is the value of $\theta$ (in degrees)?

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