Solving the system $\sum \sin = \sum \cos = 0$. Can we solve the system of equations:
$$\sin \alpha + \sin \beta + \sin \gamma = 0$$
$$\cos \alpha + \cos \beta + \cos \gamma = 0$$
?
(i.e. find the possible values of $\alpha, \beta, \gamma$)

More of a comment and less of an answer!

Well, your information seems to tell us that,
$$\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\alpha-\gamma)=-\dfrac{3}{2}$$ 
(To see this, square both equalities and add.)
I don't see any other obvious thing, you can do with these equations. If any thought plops up, I will type it in here. 
