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  4. The value of sin (9 π /14)sin ⁡ (11 π /14)sin ⁡ (13 π /14) is equal to

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Q. The value of $sin \frac{9 \pi }{14}sin ⁡ \frac{11 \pi }{14}sin ⁡ \frac{13 \pi }{14}$ is equal to

NTA AbhyasNTA Abhyas 2022

Solution:

$sin \frac{9 \pi }{14}sin ⁡ \frac{11 \pi }{14}sin ⁡ \frac{13 \pi }{14}=sin ⁡ \left(\frac{\pi }{2} + \frac{\pi }{7}\right)sin ⁡ \left(\frac{\pi }{2} + \frac{2 \pi }{7}\right)sin ⁡ \left(\frac{\pi }{2} + \frac{3 \pi }{7}\right)$
$=cos \frac{\pi }{7}cos ⁡ \frac{2 \pi }{7}cos ⁡ \frac{3 \pi }{7}$
$=cos \frac{\pi }{7}cos ⁡ \frac{2 \pi }{7}cos ⁡ \left(\pi - \frac{4 \pi }{7}\right)$
$=-cos \frac{\pi }{7}cos ⁡ \frac{2 \pi }{7}cos ⁡ \frac{4 \pi }{7}$
$=\frac{- sin \frac{8 \pi }{7}}{8 sin ⁡ \frac{\pi }{7}}=\frac{- sin ⁡ \left(\pi + \frac{\pi }{7}\right)}{8 sin ⁡ \frac{\pi }{7}}$
$=\frac{1}{8}$