Quote Originally Posted by PhilosopherStoned View Post
Xei, do you happen to know a good way to derive the formulas for e.g., cos(a + b), that doesn't use ei(a + b) or rotation matrices? Those are the only two I know and it would be good to know one that doesn't require complex multiplication or matrix multiplication.
Yes; I know a geometrical argument for sin(A+B) which is the simplest one. All variations you can then get using cos(x) = sin(pi/2 - x), sin(-x) = -sin(x), and cos(-x) = cos(x).

All the other stuff like trig(A)trig(B) and trig(A) + trig(B) also follows from that one geometrical argument.

Basically you have two right angled triangles one on top of each other; angles A and B at the origin, the bottom triangle's adjacent horizontal, and the top triangle's adjacent the same as the bottom triangle's hypotenuse. You give the top triangle's hypotenuse a length of 1 and the rest follows.

The form of a rotation matrix is usually deduced using the addition formulae so I don't think you can do it that way round. The same can apply to the Euler's identity way round, but that can also be deduced from Taylor series so that's fine. You could also do it straight from the Taylor series though it'd be a bit ugly.