Complex Phobia 5

Geometry Level 5

cos ⁡ x + cos ⁡ y + cos ⁡ z = sin ⁡ x + sin ⁡ y + sin ⁡ z = 0 \cos{x} + \cos{y }+ \cos{z} = \sin{x} + \sin{y} + \sin{z} = 0

Find : sin ⁡ 3 x + sin ⁡ 3 y + sin ⁡ 3 z \sin{3x} + \sin{3y} + \sin{3z} .

1 1 0 0 − 1 -1 3 sin ⁡ ( x + y + z ) 3\sin{(x+y+z)} 3 cot ⁡ ( x + y + z ) 3\cot{(x+y+z)}

Krishna Shankar by Krishna Shankar , 23, India

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1 solution

Prakhar Bindal
Dec 15, 2016

Consider three complex numbers

z1 = cosx+isinx

z2 = cosy+isiny

z3 = cosz+isinz

Its easy to observe that z1+z2+z3 = 0

Therefore

z1^3+z2^3+z3^3 = 3z1z2z3

Comparing imaginary parts and using de moivre's theorem we get required result

5 Helpful 0 Interesting 0 Brilliant 0 Confused

i got it wrong due to that wrong identity ! :(

A Former Brilliant Member - 4 years, 5 months ago

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