Solving the system $cos x+cos y+cos z=frac32sqrt3$, $sin x+sin y+sin z=frac32$

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 $2pi$.

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

lab bhattacharjee · Answer

Hint
$$(cos x+cos y+cos z)^2+(sin x+sin y+sin z)^2=?$$
$$impliescos(x-y)+cos(y-z)+cos(z-x)=3$$
As for $A,cos Ale1$
each of the cosine ratio will be $$=1$$

Angina Seng · Answer

We can combine these equations to state
$$e^{ix}+e^{iy}+e^{iz}=3frac{sqrt3+i}{2}.$$
But
$$|e^{ix}|+|e^{iy}|+|e^{iz}|=3=left|3frac{sqrt3+i}{2}right|
=|e^{ix}+e^{iy}+e^{iz}|.$$
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.

Solving the system $cos x+cos y+cos z=frac32sqrt3$, $sin x+sin y+sin z=frac32$

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