$\sin \frac{2 \pi}{5}+\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}+\sin \frac{8 \pi}{5}$ is equal to

$\sin \frac{2 \pi}{5}+\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}+\sin \frac{8 \pi}{5}$ is equal to
  1. 0
  2. 1
  3. $\frac{\sqrt{2}}{2}$
  4. $\frac{1}{2}$

Solution

$\sin \frac{2 \pi}{5}+\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}+\sin \frac{8 \pi}{5}$ $=\left(\sin \frac{2 \pi}{5}+\sin \frac{8 \pi}{5}\right)+\left(\sin \frac{4 \pi}{5}+\sin \frac{6 \pi}{5}\right)$ $ \begin{aligned} & =2 \sin \left(\frac{10 \pi}{5 \times 2}\right) \cos \left(\frac{-6 \pi}{5 \times 2}\right)+2 \sin \left(\frac{10 \pi}{5 \times 2}\right) \cos \left(\frac{-2 \pi}{5 \times 2}\right) \\ & =2 \sin \pi \cos \left(-\frac{3 \pi}{5}\right)+2 \sin \pi \cos \left(\frac{-2 \pi}{10}\right)[\because \sin \pi=0] \\ & =0 \end{aligned} $

Asked in: AP EAMCET 2021 (23 Aug Shift 1)

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