Trigonometry!

Since 2 sin(alpha) cos(alpha) = sin(2*alpha), we can rewrite the equation:

cos(2 alpha) = sin(2 alpha), or, dividing:

tan(2*alpha) = 1.

There are solutions where the tangent is positive (in the first and third quadrant):

2 alpha = 45 + 180 n (n can be any integer) or

2 alpha = 225 + 180 n (n can be any integer).

Dividing by two shows all solutions are of the form:

alpha = 22.5 + 90*n or

alpha = 112.5 + 90*n (n can be any integer).

Since we are restricted to the given interval, we find:

alpha = 22.5, 112.5, 202.5, or 292.5

Summing these four possibilities gives 630

$\begin{aligned} \cos 2\alpha & = 2 \sin \alpha \cos \alpha = \sin 2 \alpha \\ \tan 2 \alpha & = 1 \\ 2 \alpha & = 45^\circ, 225^\circ, 405^\circ, 585^\circ \\ \implies \alpha & = 22.5^\circ, 112.5^\circ, 202.5^\circ, 292.5^\circ \end{aligned}$

Therefore the answer is $22.5 + 112.5 + 202.5 + 292.5 = \boxed{630}$ .