I was wondering about this the other day, and it is something that was left over in my head from a thread on Euler's identity from a few weeks ago. It's a bit hard to state, but I'll try to be clear. How do you show the relationship between derivatives of sine and cosine? Now, obviously, with euler's identity and taylor series, we can show that you can break the real and imaginary parts of e^ix into the taylor series for sine and cosine and then show how they relate to the derivatives of e^ix. But how do we connect the taylor series of sine and cosine with the geometric significance of these functions? I have a feeling there is a certain set of geometric properties(*) to which there exists a unique set of functions which provably correspond to the real and imaginary parts of e^ix, but I don't quite know which set of properties that would be. (* I'm thinking Pythagorean theorem or something cos^2 x + sin^2 x = 1) I'm thinking that it should also be provable without the use of imaginary numbers, as well, as I'm pretty sure their widespread use came after study of trigonometric functions in calculus.