Question

$\sin\, \frac{\pi}{n} + \sin\, \frac{3\pi}{n} + \sin\, \frac{5\pi}{n} + ....$ upto $n$ terms is equal to

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

Answer: (D)

$\sin \frac{\pi}{n}+\sin \frac{3 \pi}{n}+\sin \frac{5 \pi}{n}+...$ upto $n$ terms
$=\frac{\sin \frac{n \cdot 2 \pi}{2 n}}{\sin \frac{2 \pi}{2 n}} \cdot \sin \left(\frac{2 \cdot \frac{\pi}{n}+(n-1) 2 \frac{\pi}{n}}{2}\right)$
$=\frac{\sin \pi}{\sin \frac{\pi}{n}} \cdot \sin \left(\frac{2 \pi+2 n \pi-2 \pi}{2 n}\right)$
$=\frac{\sin ^{2} \pi}{\sin \frac{\pi}{n}}=0$