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)

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