# Solving the system $\cos x+\cos y+\cos z=\frac32\sqrt3$, $\sin x+\sin y+\sin z=\frac32$

Suppose we have \begin{align} \cos x + \cos y + \cos z &= \frac{3}{2}\sqrt{3} \\[4pt] \sin x + \sin y + \sin z &= \frac{3}{2} \end{align}

How can we solve for $$x$$, $$y$$ and $$z$$?

According to Wolfram Alpha, the values of $$x, y, z$$ must be the same i.e. $$\pi/6$$ modulo $$2\pi$$.

How do we solve the equations analytically?

What I am able to prove. I am able to show that two out of three variables $$x,y, z$$ must be equal. This I can do by reformulating the problem as "maximize $$\sin x$$ subject to the above constraints." and doing Lagrange optimization. I am sure there must be a simpler way.

Problem source: From CMI Entrance 2010 paper

$$(\cos x+\cos y+\cos z)^2+(\sin x+\sin y+\sin z)^2=?$$
$$\implies\cos(x-y)+\cos(y-z)+\cos(z-x)=3$$
As for $$A,\cos A\le1$$
each of the cosine ratio will be $$=1$$
We can combine these equations to state $$e^{ix}+e^{iy}+e^{iz}=3\frac{\sqrt3+i}{2}.$$ But $$|e^{ix}|+|e^{iy}|+|e^{iz}|=3=\left|3\frac{\sqrt3+i}{2}\right| =|e^{ix}+e^{iy}+e^{iz}|.$$ So equality holds in the triangle inequality; if $$|u|+|v|+|w|=|u+v+w|$$ then $$u$$, $$v$$ and $$w$$ are non-negative multiples of the same complex number. So $$e^{ix}=e^{iy}=e^{iz}=(\sqrt3+i)/2$$ etc.