sin frac{pi}{5} + sin frac{2pi}{5} + sin frac | vedclass

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(B) We need to evaluate $S = \sin \frac{\pi}{5} + \sin \frac{2\pi}{5} + \sin \frac{3\pi}{5} + \sin \frac{4\pi}{5}$.
Note that $\sin \frac{4\pi}{5} =

Vedclass · Accepted Answer

$ \sqrt{5 + 2\sqrt{5}}$

Vedclass · Answer

(B) We need to evaluate $S = \sin \frac{\pi}{5} + \sin \frac{2\pi}{5} + \sin \frac{3\pi}{5} + \sin \frac{4\pi}{5}$.
Note that $\sin \frac{4\pi}{5} = \sin(\pi - \frac{\pi}{5}) = \sin \frac{\pi}{5}$ and $\sin \frac{3\pi}{5} = \sin(\pi - \frac{2\pi}{5}) = \sin \frac{2\pi}{5}$.
So,$S = 2(\sin \frac{\pi}{5} + \sin \frac{2\pi}{5})$.
Using the values $\sin \frac{\pi}{5} = \frac{\sqrt{10 - 2\sqrt{5}}}{4}$ and $\sin \frac{2\pi}{5} = \frac{\sqrt{10 + 2\sqrt{5}}}{4}$,
$S = 2 \left( \frac{\sqrt{10 - 2\sqrt{5}} + \sqrt{10 + 2\sqrt{5}}}{4} \right) = \frac{\sqrt{10 - 2\sqrt{5}} + \sqrt{10 + 2\sqrt{5}}}{2}$.
Squaring $S$: $S^2 = \frac{1}{4} (10 - 2\sqrt{5} + 10 + 2\sqrt{5} + 2\sqrt{(10 - 2\sqrt{5})(10 + 2\sqrt{5})}) = \frac{1}{4} (20 + 2\sqrt{100 - 20}) = \frac{1}{4} (20 + 2\sqrt{80}) = \frac{1}{4} (20 + 8\sqrt{5}) = 5 + 2\sqrt{5}$.
Thus,$S = \sqrt{5 + 2\sqrt{5}}$.

$\sin \frac{\pi}{5} + \sin \frac{2\pi}{5} + \sin \frac{3\pi}{5} + \sin \frac{4\pi}{5} =$

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