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 $\alpha, \beta, \gamma$)

Developing on Gerenuk's answer, you could consider the complex numbers 
$$ z_1=\cos \alpha+i\sin \alpha,\ z_2=\cos \beta+i\sin\beta,\ z_3=\cos \gamma+i\sin \gamma$$
Then you know that $z_1,z_2,z_3$ are on the unit circle, and the centroid of the triangle formed by the points of afixes $z_i$ is of afix $\frac{z_1+z_2+z_3}{3}=0$. From classical geometry, we can see that if the centroid of a triangle is the same as the center of the circumscribed circle, then the triangle is equilateral. This proves that $\alpha,\beta,\gamma$ are of the form $\theta,\theta+\frac{2\pi}{3},\theta+\frac{4\pi}{3}$.
