કિંમત શોધો: cos frac{pi}{7} cos frac{2 pi}{7} | vedclass

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Question

(C) ધારો કે $P = \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \cos \frac{\pi}{14} \cos \frac{3 \pi}{14} \cos \frac{5 \pi}{14}$.નિત્યસમ

Vedclass · Accepted Answer

$\frac{1}{32}\left[\sin \frac{2 \pi}{7}+\sin \frac{3 \pi}{7}-\sin \frac{\pi}{7}\right]$

Vedclass · Answer

(C) ધારો કે $P = \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \cos \frac{\pi}{14} \cos \frac{3 \pi}{14} \cos \frac{5 \pi}{14}$.નિત્યસમ $\cos 	heta = \sin(\frac{\pi}{2} - 	heta)$ નો ઉપયોગ કરતા,આપણને મળે છે $\cos \frac{\pi}{14} = \sin \frac{3 \pi}{7}$,$\cos \frac{3 \pi}{14} = \sin \frac{2 \pi}{7}$,અને $\cos \frac{5 \pi}{14} = \sin \frac{\pi}{7}$.આ કિંમતો મૂકતા,$P = \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \sin \frac{\pi}{7} \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}$.ગોઠવતા,$P = (\sin \frac{\pi}{7} \cos \frac{\pi}{7}) (\sin \frac{2 \pi}{7} \cos \frac{2 \pi}{7}) (\sin \frac{3 \pi}{7} \cos \frac{3 \pi}{7})$.$2 \sin 	heta \cos 	heta = \sin 2 	heta$ નો ઉપયોગ કરતા,$P = \frac{1}{8} (\sin \frac{2 \pi}{7} \sin \frac{4 \pi}{7} \sin \frac{6 \pi}{7})$.$\sin \frac{6 \pi}{7} = \sin \frac{\pi}{7}$ અને $\sin \frac{4 \pi}{7} = \sin \frac{3 \pi}{7}$ હોવાથી,$P = \frac{1}{8} \sin \frac{\pi}{7} \sin \frac{2 \pi}{7} \sin \frac{3 \pi}{7}$.$2 \sin A \sin B = \cos(A-B) - \cos(A+B)$ નો ઉપયોગ કરતા,સાદું રૂપ આપતા જવાબ $\frac{1}{32} [\sin \frac{2 \pi}{7} + \sin \frac{3 \pi}{7} - \sin \frac{\pi}{7}]$ મળે છે.

કિંમત શોધો: $\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{3 \pi}{7} \cos \frac{\pi}{14} \cos \frac{3 \pi}{14} \cos \frac{5 \pi}{14}$

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