trigonometry - Sine of the sum of angles

I need to prove the formula for the sine of the sum

$$\sin(\alpha+\beta) = \sin(\alpha)\cos(\beta) + \sin(\beta)\cos(\alpha)$$
I already know how to prove it when $\alpha,\beta\geq 0$ and $\alpha+\beta <\pi/2$. How can I extend it to any pair of angles? The definition of sine and cosine that I am using is the length of the $y$-axis and the $x$-axis respectively when you intersect the circle of radius 1, but I can't use analytic geometry. Also I can't use complex numbers multiplication. Only relations like $\sin(\alpha +\pi/2) = \cos(\alpha)$.