Company with sines

Geometry Level 2

Find: sin ⁡ θ + sin ⁡ ( π + θ ) + sin ⁡ ( 2 π + θ ) + sin ⁡ ( 3 π + θ ) + . . . . \sin \theta+\sin (\pi+\theta)+\sin(2\pi+\theta)+\sin(3\pi+\theta)+.... upto 2012 terms.

0 1006 sin ⁡ θ \sin \theta 1

Ayush Sharma by Ayush Sharma , 20, India

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2 solutions

Deepak Kumar
Jul 27, 2015

Hint:The last term will be sin(2011pi+x) and clearly each pair from beginning is 0 and hence all 1006 pairs sum up to 0

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Omkar Kulkarni
Jul 25, 2015

sin ⁡ ( 2 n π + θ ) = sin ⁡ θ \sin(2n\pi+\theta)=\sin\theta

sin ⁡ ( ( 2 n + 1 ) π + θ ) = sin ⁡ ( π + θ ) = − sin ⁡ θ \sin((2n+1)\pi+\theta)=\sin(\pi+\theta)=-\sin\theta

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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