Question

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

• A. $\frac{1}{64}$
• B. $-\frac{1}{64}$
• C. $\frac{1}{8}$
• D. $-\frac{1}{8}$

Answer 1

Answer: (C)

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