# Sum of Sines or Cosines of n Angles in A.P

## Sum of Sines or Cosines of n Angles in A.P

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

Proof: Let S = sinα + sin(α + β) + sin(α + 2β) + . . . + sin( α + (n – 1)β)

Here angles are in A.P and common difference of angles in β

$$=\frac{\sin \frac{n\beta }{2}}{\sin \frac{\beta }{2}}\times \sin \left( \frac{First\ angle\ \ +\ \ last\ \ angle}{2} \right)$$.

multiplying both sides by 2sin(β/2), we get

2sin(β/2) S = sinα 2sin(β/2) + sin (α + β) 2sin(β/2) + sin (α + 2β) 2sin(β/2) + . . . + sin (α + (n – 1) β) 2sin(β/2)

(Since 2 sinA sinB = cos (A – B) – cos (A + B))

$$2sin\alpha \text{ sin}\left( \frac{\beta }{2} \right)=\cos \left( \alpha -\frac{\beta }{2} \right)-\cos \left( \alpha +\frac{\beta }{2} \right)$$,

$$2sin\left( \alpha +\beta \right)\text{ sin}\left( \frac{\beta }{2} \right)=\cos \left( \alpha +\beta -\frac{\beta }{2} \right)-\cos \left( \alpha +\beta +\frac{\beta }{2} \right)$$,

$$=\cos \left( \alpha +\frac{\beta }{2} \right)-\cos \left( \alpha +\frac{3\beta }{2} \right)$$,

$$2sin\left( \alpha +2\beta \right)\text{ sin}\left( \frac{\beta }{2} \right)=\cos \left( \alpha +2\beta -\frac{\beta }{2} \right)-\cos \left( \alpha +2\beta +\frac{\beta }{2} \right)$$,

$$=\cos \left( \alpha +\frac{3\beta }{2} \right)-\cos \left( \alpha +\frac{5\beta }{2} \right)$$,

From above equations,

$$2\sin \left( \frac{\beta }{2} \right)S=\cos \left( \alpha -\frac{\beta }{2} \right)-\cos \left( \alpha +\left( 2n-1 \right)\frac{\beta }{2} \right)$$,

$$2\sin \left( \frac{\beta }{2} \right)S=2\sin \left( \alpha +\left( n-1 \right)\frac{\beta }{2} \right)\sin \left( \frac{n\beta }{2} \right)$$,

$$S=\frac{\sin \left( \frac{n\beta }{2} \right)}{\sin \left( \frac{\beta }{2} \right)}\sin \left( \alpha +\left( n-1 \right)\frac{\beta }{2} \right)$$,

In the above result, replacing α by π/2 + α, we get

$$\cos \alpha +\cos \left( \alpha +\beta \right)+\cos \left( \alpha +2\beta \right)+…+\cos \left( \alpha +\left( n-1 \right)\beta \right)=\frac{\sin \frac{n\beta }{2}}{\sin \frac{\beta }{2}}\cos \left( \alpha +\left( n-1 \right)\frac{\beta }{2} \right)$$.

Hence proved.

Example: Find the value of  $$\cos \left( \frac{2\pi }{7} \right)+\cos \left( \frac{4\pi }{7} \right)+\cos \left( \frac{6\pi }{7} \right)$$.

Solution: Given that $$\cos \left( \frac{2\pi }{7} \right)+\cos \left( \frac{4\pi }{7} \right)+\cos \left( \frac{6\pi }{7} \right)$$,

$$S=\frac{\sin \left( \frac{n\beta }{2} \right)}{\sin \left( \frac{\beta }{2} \right)}\sin \left( \alpha +\left( n-1 \right)\frac{\beta }{2} \right)$$,

$$=\frac{\sin \left( 3\frac{\pi }{7} \right)}{\sin \left( \frac{\pi }{7} \right)}\cos \left( \frac{\pi }{7}+\frac{3\pi }{7} \right)$$,

$$=\frac{2\sin \left( \frac{3\pi }{7} \right)\cos \left( \frac{4\pi }{7} \right)}{2\sin \left( \frac{\pi }{7} \right)}$$,

$$=\frac{\sin \left( \frac{7\pi }{7} \right)-\sin \left( \frac{\pi }{7} \right)}{2\sin \left( \frac{\pi }{7} \right)}=-\frac{1}{2}$$.