If cos theta + cos 7theta + cos 3theta + cos | vedclass

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Question

(C) Given equation: $\cos 	heta + \cos 7	heta + \cos 3	heta + \cos 5	heta = 0$Using the sum-to-product formula $\cos C + \cos D = 2 \cos \frac{C+D

Vedclass · Accepted Answer

$\frac{n\pi}{8}$

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(C) Given equation: $\cos 	heta + \cos 7	heta + \cos 3	heta + \cos 5	heta = 0$Using the sum-to-product formula $\cos C + \cos D = 2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}$:$(\cos 7	heta + \cos 	heta) + (\cos 5	heta + \cos 3	heta) = 0$$2 \cos 4	heta \cos 3	heta + 2 \cos 4	heta \cos 	heta = 0$$2 \cos 4	heta (\cos 3	heta + \cos 	heta) = 0$Using $\cos 3	heta + \cos 	heta = 2 \cos 2	heta \cos 	heta$:$2 \cos 4	heta (2 \cos 2	heta \cos 	heta) = 0$$4 \cos 4	heta \cos 2	heta \cos 	heta = 0$This implies $\cos 4	heta = 0$ or $\cos 2	heta = 0$ or $\cos 	heta = 0$.If $\cos 4	heta = 0$,then $4	heta = (2n+1)\frac{\pi}{2} \implies 	heta = (2n+1)\frac{\pi}{8}$.If $\cos 2	heta = 0$,then $2	heta = (2n+1)\frac{\pi}{2} \implies 	heta = (2n+1)\frac{\pi}{4}$.If $\cos 	heta = 0$,then $	heta = (2n+1)\frac{\pi}{2}$.Combining these,the general solution is $	heta = \frac{n\pi}{8}$.

If $\cos \theta + \cos 7\theta + \cos 3\theta + \cos 5\theta = 0$,then $\theta$ is equal to:

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