What axioms for geometry and trigonometry would I have to chose in order to completely avoid coordinates in defining trig functions and showing the equivalence of their geometric (unit circle) and series realizations.

This is usually done by calculating arclengt in a coordinate system but I want to avoid completely any explicit or implicit use of coordinates.

Since a circle is a set of all points equidistant from a given point, it follows that it is a limit of a sequence of regular n sided polygons as n goes to infinity

How do I show that a half lenght side of one n sided poligon is the sine function of pi over n? Without in any way introducing coordinates

What way would there even be to define or calculate pi in such an axiomatic coordinate less setting