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

Question

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

Answer 1

Hint

$$(\cos x+\cos y+\cos z)^2+(\sin x+\sin y+\sin z)^2=?$$

$$\implies\cos(x-y)+\cos(y-z)+\cos(z-x)=3$$

As for $A,\cos A\le1$

each of the cosine ratio will be $$=1$$

Answer 2

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.