Question

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

• A. $\frac{1}{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $1$
• D. $-\frac{1}{2}$

Answer 1

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

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

$\to \sin \frac{\pi }{10}+\sin (\pi +\frac{3\pi }{10})$

$\to \sin \frac{\pi }{10}-\sin \frac{3\pi }{10}$

we knoe thta $\sin \frac{\pi }{10}=\sin {18}^o=-\frac{1+\sqrt{5}}{4}$

$\sin \frac{3\pi }{10}=\sin {54}^o=\frac{\sqrt{5}+1}{4}$

$\sin \frac{\pi }{10}-\sin \frac{3\pi }{10}=-\frac{1+\sqrt{5}}{4}-\frac{(\sqrt{5}-1)}{4}=\frac{-2}{4}=\frac{-1}{2}$

$D$ is correct