Question

Solve
$\sin \frac{\pi }{10}+\sin \frac{13\pi }{10}$

• A. $-\frac{1}{2}$
• B. $-\frac{1}{3}$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is A $-\frac{1}{2}$

$\sin \frac{\pi }{10}+\sin \frac{13\pi }{10}=\sin {18}^{\circ }+\sin {234}^{\circ }=\sin {18}^{\circ }+\sin ({270}^{\circ }-{36}^{\circ })=\sin {18}^{\circ }-\cos {36}^{\circ }$

As we know that $\sin {18}^{\circ }=\frac{\sqrt{5}-1}{4},\cos {36}^{\circ }=\frac{\sqrt{5}+1}{4}$

So $\sin \frac{\pi }{10}+\sin \frac{13\pi }{10}=\frac{\sqrt{5}-1}{4}-\frac{\sqrt{5}+1}{4}=\frac{-1-1}{4}=\frac{-1}{2}$