This is something I have always had issues with. Generally, three approaches are used:
- The geometric path: this follows the standard way how you would introduce these functions in school. The problem is that it is almost impossible to do this in a rigorous way and you have to introduce difficult concepts (like arc-length) what you would normally do later on.
- The power-series road: the advantage is that this is quick, elegant, and you can prove all the properties in a quick and not too painful way. The problem is that for many students this is ad-hoc, the definition is unmotivated, and there is no connection with the geometric meaning.
- The functional equation road: to define them through functional identities and continuity, and then prove properties. This is halfway motivated because at least addition formulae can be assumed from school, but, as I had it this way as I was a student, I have some reservations because I remember that this approach left some odd feeling.
If your students had a calculus class, then clearly the second approach can be used and this is what I do with those students who have this class after a calculus class.
But generally, in my university, it is common that first year students start their studies with a rigorous analysis course. Which approach do you consider as the best in this case, where I cannot assume much familiarity with calculus knowledge and also do not want to wait till power series appear?