Sineful mistake

A common misconception of students is to claim that $\sin x^\circ + \sin y^\circ = \sin [ (x+y)^ \circ],$ just because $x + y = (x+y)$. There are $N$ ordered pairs of integers $(x, y)$, each of which are between 0 and 360 (inclusive), which satisfy the equation $\sin x^\circ + \sin y^\circ = \sin [(x+y)^ \circ].$

What are the last three digits of $N$?

Details and assumptions

Notice that the equations are given in degrees, not radians.