Question

The value of $sin\frac{\pi }{10}sin\frac{13\pi }{10}$ is

• A. $\frac{1}{2}$
• B. $-\frac{1}{2}$
• C. $-\frac{1}{4}$
• D. $1$

Answer 1

Answer: (C)

$sin\frac{\pi }{10}sin\frac{13\pi }{10}=sin\frac{\pi }{10}sin\left(\pi +\frac{3\pi }{10}\right)$
$=-sin\frac{\pi }{10}sin\frac{3\pi }{10}$
$=-\frac{1}{2}\left[2sin\frac{\pi }{10}sin\frac{3\pi }{10}\right]$
$=-\frac{1}{2}\left[cos\left(\frac{\pi }{10}-\frac{3\pi }{10}\right)-cos\left(\frac{\pi }{10}+\frac{3\pi }{10}\right)\right]$
$=-\frac{1}{2}\left[cos\left(-\frac{2\pi }{10}\right)-cos\left(\frac{4\pi }{10}\right)\right]$
$=-\frac{1}{2}\left[cos\frac{\pi }{5}-cos\frac{2\pi }{5}\right]$
$=-\frac{1}{2}\left[cos36^{\circ }-cos72^{\circ }\right]$
$=-\frac{1}{2}\left[\frac{\sqrt{5}+1}{4}-\frac{\sqrt{5}-1}{4}\right]$
$=-\frac{1}{2}\left[\frac{2}{4}\right]=-\frac{1}{4}$