કિંમત શોધો: sin ^4 frac{pi}{8} + sin ^4 frac{ | vedclass

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(C) આપેલ પદાવલિ: $S = \sin ^4 \frac{\pi}{8} + \sin ^4 \frac{3\pi}{8} + \sin ^4 \frac{5\pi}{8} + \sin ^4 \frac{7\pi}{8}$ $\sin(\pi - 	heta) = \sin 	h

Vedclass · Accepted Answer

$\frac{3}{2}$

Vedclass · Answer

(C) આપેલ પદાવલિ: $S = \sin ^4 \frac{\pi}{8} + \sin ^4 \frac{3\pi}{8} + \sin ^4 \frac{5\pi}{8} + \sin ^4 \frac{7\pi}{8}$ $\sin(\pi - 	heta) = \sin 	heta$ હોવાથી,$\sin \frac{7\pi}{8} = \sin \frac{\pi}{8}$ અને $\sin \frac{5\pi}{8} = \sin \frac{3\pi}{8}$ થાય. તેથી,$S = 2 \left( \sin ^4 \frac{\pi}{8} + \sin ^4 \frac{3\pi}{8} ight)$. નિત્યસમ $\sin^2 	heta = \frac{1 - \cos 2	heta}{2}$ નો ઉપયોગ કરતા,$\sin^4 	heta = \frac{1}{4} (1 - 2\cos 2	heta + \cos^2 2	heta)$. $S = 2 \left[ \frac{1}{4} (1 - 2\cos \frac{\pi}{4} + \cos^2 \frac{\pi}{4}) + \frac{1}{4} (1 - 2\cos \frac{3\pi}{4} + \cos^2 \frac{3\pi}{4}) ight]$ $S = \frac{1}{2} \left[ 1 - 2(\frac{1}{\sqrt{2}}) + \frac{1}{2} + 1 - 2(-\frac{1}{\sqrt{2}}) + \frac{1}{2} ight]$ $S = \frac{1}{2} \left[ 1 - \sqrt{2} + \frac{1}{2} + 1 + \sqrt{2} + \frac{1}{2} ight] = \frac{1}{2} [3] = \frac{3}{2}$.

કિંમત શોધો: $\sin ^4 \frac{\pi}{8} + \sin ^4 \frac{3\pi}{8} + \sin ^4 \frac{5\pi}{8} + \sin ^4 \frac{7\pi}{8} = $

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