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  4. The value of sin (π/18)+sin (π/9)+sin (2π /9)+sin (5π/18) is given by

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Q. The value of $sin \frac{\pi}{18}+sin \frac{\pi}{9}+sin \frac{2\pi }{9}+sin \frac{5\pi}{18}$ is given by

Trigonometric Functions

Solution:

We have,
$sin \frac{\pi}{18}+sin \frac{\pi}{9}+sin \frac{2\pi }{9}+sin \frac{5\pi}{18}$
$=sin\left(\pi-\frac{17\pi}{18}\right)+sin\left(\pi -\frac{8\pi }{9}\right)+sin \frac{2\pi}{9}+sin \frac{5\pi}{18}$
$=sin \frac{17\pi}{18}+sin \frac{8\pi }{9}+sin \frac{2\pi }{9}+sin \frac{5\pi}{18}$
$=\left(sin \frac{17\pi }{18}+sin \frac{5\pi }{18}\right)+\left(sin \frac{8\pi }{9}+sin \frac{2\pi }{9}\right)$
$=2\,sin\left(\frac{11\pi}{18}\right)cos \frac{\pi}{3}+2\,sin \frac{5\pi}{9} cos \frac{\pi}{3}$
$=sin \frac{11\pi }{18}+sin \frac{5\pi }{9}$
$=sin\left(\pi-\frac{7\pi}{18}\right)+sin\left(\pi-\frac{4\pi}{9}\right)$
$=sin \frac{7\pi }{18}+sin \frac{4\pi }{9}$