Sineful mistake

Geometry Level 5

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 \(N\)?