Question

If S = sin$\frac{\pi }{n}$ + sin$\frac{3\pi }{n}$ + sin$\frac{5\pi }{n}$+..........n terms.

Find the value nS

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Answer 1

Given series S = sin$\frac{\pi }{n}$ + sin$\frac{3\pi }{n}$ + sin$\frac{5\pi }{n}$+ ..........sin$(\alpha +(n-1\left)\beta \right)$.

Comparing the given series with standard sine series

sin$\alpha$ + sin($\alpha +\beta$) + sin($\alpha +2\beta$) + sin($\alpha +3\beta$) + ..............

$\alpha =\frac{\pi }{n}$

$\beta =\frac{2\pi }{n}$

Sum of the sine series = $\frac{sin\frac{n\beta }{2}}{sin\frac{\beta }{2}}$ sin$(\alpha +\frac{(n-1)\beta }{2})$

= $\frac{sin\left(\frac{n\times 2\pi }{2\times n}\right)}{sin\left(\frac{2\pi }{2\times n}\right)}sin(\frac{\pi }{n}+\frac{(n-1)2\pi }{2.n})$

= $\frac{sin\pi }{sin\left(\frac{\pi }{n}\right)}\times sin(\frac{\pi }{n}+\frac{(n-1)\pi }{n})$

= $\frac{0}{sin\left(\frac{\pi }{n}\right)}.sin(\frac{\pi }{n}+\frac{(n-1)\pi }{n})$

= 0

n.S = n $\times$ 0 = 0