They’re a lot easier with Euler’s formula.
Why do we care about $sin(a + b)$ – why not just have c = a + b, and do $sin©$ ?
- Sometimes we want to see how the individual parts show up in the final. Don’t mix them up, keep their contributions separate.
- Realistically, this is used for Calculus, where we might have an integral (which is some combination) and want to get the individual parts back out.
Use Euler’s Formula to see sin(a + b) as getting the “curtain height” after doing two rotations:
- from flat (0 degrees/radians) to $a$
- and then rotate again from $a$ to $a + b$.