Complex Phobia 3

Geometry Level 4

If $\cos{x} + \cos{y }+ \cos{z} = \sin{x} + \sin{y} + \sin{z} = 0$

Find : $\sin^2 {x} + \sin^2{y} + \sin^2{z}$

0

\dfrac{3}{2}

\dfrac{2}{3}

1

\dfrac{1}{2}

1 solution

Aryan Goyat
Dec 20, 2016

Consider three complex numbers

z1 = cosx+isinx

z2 = cosy+isiny

z3 = cosz+isinz

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

(z1)^2+(z2)^2+(z3)^2=(-2z1 z2-2z2 z3-2z3 z1) =-2z1 z2*z3(cosx+cosy+cosz-i(sin(x)+sin(y)+sin(z)))

implies real part of (z1)^2+(z2)^2+(z3)^2=0 next its easy