1. Tardigrade
  2. Question
  3. Mathematics
  4. sin (π/14) sin (3π/14) sin (5π/14) sin (7π/14) is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. $sin \frac{\pi}{14}\, sin \frac{3\pi}{14} \,sin \frac{5\pi}{14} \,sin \frac{7\pi}{14}$ is equal to

UPSEEUPSEE 2010

Solution:

$\sin \frac{\pi}{14} \sin \frac{3 \pi}{14} \sin \frac{5 \pi}{14} \sin \frac{7 \pi}{14}$
$=\sin \left(\frac{\pi}{2}-\frac{3 \pi}{7}\right) \sin \left(\frac{\pi}{2}-\frac{2 \pi}{7}\right) \sin \left(\frac{\pi}{2}-\frac{\pi}{7}\right) \sin \frac{\pi}{2}$
$=\cos \frac{3 \pi}{7} \cos \frac{2 \pi}{7} \cos \frac{\pi}{7}$
$=-\cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$
$=-\frac{\sin \left(2^{3} \pi / 7\right)}{2^{3} \sin \pi / 7}=-\frac{\sin 8 \pi / 7}{8 \sin \pi / 7}=\frac{1}{8}$