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Geometry Level 4

Find the value of the sum c o s A + c o s ( A + 2 π 1729 ) + c o s ( A + 4 π 1729 ) + . . . . t o 1729 t e r m s cosA+cos\left( A+\frac{2\pi}{1729} \right) +cos\left( A+\frac{4\pi}{1729} \right) +.... to \ 1729 \ terms

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Sandeep Bhardwaj by Sandeep Bhardwaj , 28, India

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1 solution

Pranjal Jain
Dec 12, 2014

cos ⁡ x + cos ⁡ ( x + y ) + cos ⁡ ( x + 2 y ) + . . . + cos ⁡ ( x + ( n − 1 ) y ) = sin ⁡ n y 2 sin ⁡ y 2 × cos ⁡ ( x + ( n − 1 ) y 2 ) \cos x+\cos (x+y)+\cos (x+2y)+...+\cos (x+(n-1)y)=\frac{\sin\frac{ny}{2}}{\sin\frac{y}{2}}×\cos(x+\frac{(n-1)y}{2})

Substitute x = A , y = 2 π 1729 , n = 1729 x=A,\ y=\frac{2\pi}{1729},\ n=1729 ,

S = sin ⁡ 1729 × 2 π 1729 2 sin ⁡ 2 π 1729 2 × cos ⁡ ( A + 1728 × 2 π 1729 2 ) S=\dfrac{\sin\dfrac{1729×\frac{2\pi}{1729}}{2}}{\sin\dfrac{\frac{2\pi}{1729}}{2}}×\cos(A+\frac{1728×\frac{2\pi}{1729}}{2}) S = 0 S=0 as sin ⁡ π = 0 \sin\pi=0

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