The value of sin frac{pi}{14} sin frac{3pi}{1 | vedclass

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(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{

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,$\sin \frac{7\pi}{14} = \sin \frac{\pi}{2} = 1$.Thus,$P = \left( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} ight)^2 	imes 1$.Using the identity $\prod_{k=1}^{n} \sin \frac{k\pi}{2n+1} = \frac{\sqrt{2n+1}}{2^n}$,for $2n+1 = 7$ (where $n=3$),we have $\sin \frac{\pi}{7} \sin \frac{2\pi}{7} \sin \frac{3\pi}{7} = \frac{\sqrt{7}}{8}$.However,for the given expression,using the product formula $\prod_{k=1}^{n} \sin \frac{k\pi}{2n+1} = \frac{\sqrt{2n+1}}{2^n}$,the value evaluates to $\frac{1}{64}$.

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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