List of trigonometric identities

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 $a + b$.

The motivation we’re usually given is that those are the only values given in the question; a bit lacking in a sense of purpose, isn’t it?

I agree. Originally trig identities were also used as a type of shortcut for multiplication (similar to log tables).

I think it’s silly to pretend the identities are “useful for computation” when the majority of the time we are just going to plug sin(a) and sin(b) into a calculator to find their values. In my head the primary reason is keeping track of the contribution of a part a whole.