This might seem like a really basic question that one might cover in gr 9 or 10 but instead my friend and I were discussing it now, when he just got his degree and I'm a credit away from mine: why on earth are there both sine and cosine functions when simply one would do? Either can be expressed in terms of the other with a phase shift of π/2 and there weren't any identities we could think of that would necessitate an entirely new function to express that. My only thought was that maybe standard sine and cosine functions weren't originally posed as themselves, instead having been derived using complex exponentials -- the +/- between the terms could warrant the use of separate functions for a notational convenience. The other thought I had was that maybe they weren't posed first, and were derived from their hyperbolic analogs. While neither of my suggestions seem likely, we couldn't think of any other reason that we would need to define two functions instead of one function of different arguments. If anyone has any insight I'd love to hear it. Thanks!