How to compute sum of angles from sine and cossine?

Question

What is the fastest way to know if the a + b > pi, having only sine and cossine of both a and b? I am hoping to not have to use arccos or arcsin.

Edit: forgot to mention that a and b are in range [0, pi).

Answer 1

With the restriction of the angles domain to [0, pi) that's straightforward. You just have to check if sin(a+b) < 0 and if you already know sin and cos of a and b, then:

sin(a + b) = sin(a)*cos(b) + sin(b)*cos(a)

is what you want.