Question

If $\theta$ is the exterior angle of a regular polygon of n sides and $\alpha$ is constant, then find the value of sin $\alpha$ + sin $(\alpha +\theta )$ + sin $(\alpha +2\theta )$ . . . . . up to n terms

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Answer 1

Since $\theta$ is the exterior angle of a regular polygon, each exterior angle of n sided regular polygon = $\frac{{360}^{\circ }}{n}$.

Sum of the above given sine series.

= $\frac{sin\frac{n\theta }{2}}{sin\frac{\theta }{2}}.sin(\alpha +\frac{(n-1)\theta }{2})$

$\theta$ = $\frac{{360}^{\circ }}{n}$

Sum = $\frac{sin\frac{n\times 360}{n.2}}{sin\frac{360}{n.2}}\times sin(\alpha +\frac{(n-1)\times 360}{2})$

= $\frac{sin{180}^{\circ }}{sin\frac{{180}^{\circ }}{n}}.sin(\alpha +(n-1\left){180}^{\circ }\right)$

We know, sin ${180}^{\circ }$ = 0

= $\frac{0}{sin\frac{{180}^{\circ }}{n}}\times sin(\alpha +(n-1)\times {180}^{\circ })$

= 0