The value of sin frac{pi }{14} sin frac{3pi } | vedclass

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Question

(D) Let $P = \sin \frac{\pi }{14} \sin \frac{3\pi }{14} \sin \frac{5\pi }{14} \sin \frac{7\pi }{14} \sin \frac{9\pi }{14} \sin \frac{11\pi }{14} \sin

Vedclass · Accepted Answer

$\frac{1}{64}$

Vedclass · Answer

(D) Let $P = \sin \frac{\pi }{14} \sin \frac{3\pi }{14} \sin \frac{5\pi }{14} \sin \frac{7\pi }{14} \sin \frac{9\pi }{14} \sin \frac{11\pi }{14} \sin \frac{13\pi }{14}$.Using the property $\sin(\pi - 	heta) = \sin 	heta$,we have:$\sin \frac{13\pi }{14} = \sin \frac{\pi }{14}$,$\sin \frac{11\pi }{14} = \sin \frac{3\pi }{14}$,and $\sin \frac{9\pi }{14} = \sin \frac{5\pi }{14}$.Also,$\sin \frac{7\pi }{14} = \sin \frac{\pi }{2} = 1$.Substituting these,we get:$P = \left( \sin \frac{\pi }{14} \sin \frac{3\pi }{14} \sin \frac{5\pi }{14} ight)^2 	imes 1$.Using the identity $\prod_{k=1}^{n} \sin \frac{k\pi}{2n+1} = \frac{\sqrt{2n+1}}{2^n}$,for $n=7$,$2n+1=15$ (not applicable directly). Alternatively,using $\sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} = \frac{1}{8}$,we get:$P = \left( \frac{1}{8} ight)^2 	imes 1 = \frac{1}{64}$.

The value of $\sin \frac{\pi }{14} \sin \frac{3\pi }{14} \sin \frac{5\pi }{14} \sin \frac{7\pi }{14} \sin \frac{9\pi }{14} \sin \frac{11\pi }{14} \sin \frac{13\pi }{14}$ is equal to

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