Take the 2-minute tour ×  Sometimes, there are many ways to give
defintion to a mathmatical principle, for example the natural
base logarithm.  How about sine and cosine?   share|improve this
question A lot. I think this question, as written, is too vague.
–  Qiaochu Yuan Aug 26 '10 at 4:48 (I think I have figured out
what bothers me about this question. This question presupposes
that the ways of defining sine and cosine form a set. Really
they form a category in which all of the objects are
isomorphic...) –  Qiaochu Yuan Aug 26 '10 at 5:03 @Qiaochu Yuan:
I'm not sure if your previous comment was meant absolutely
serious. If so, how do you define the morphism set between two
definitions of sine? –  Rasmus Aug 26 '10 at 9:27 @Rasmus: it
was a metaphor. I just mean that one shouldn't think of the
different definitions as unrelated, as one can get from any one
to any other one. I also meant that it doesn't make sense to ask
about the cardinality of the objects, since this is an "evil"
question; cardinality is no