Let's say I want to study real analysis in a self-contained fashion by starting with the axioms of the real numbers. This is because I already know the basic concepts of set theory and high school calculus.
I have already defined the concepts of the exponential and logarithmic functions by using the upper bound property. I also defined the decimal representation of numbers, again by using the upper bound property.
My problem is how to define trigonometric functions before I even start the study of derivatives, integrals or power series. In this I'm thinking of a definition analytically, meaning without appealing to geometry other than intuitively.
The reason I'm trying to do this is because in the study of derivatives or integrals they use a lot of examples including real functions without they being previously defined, which makes me feel a little uncomfortable. One way to get around this is probably by just thinking of those functions axiomatically, meaning they will be defined at some point later and meanwhile they have certain properties that I can use to find derivatives and integrals, but still I wanted to know the elementary way to accomplish this.