The solution set to equations of the form X^2 + Y^2 = R^2?

The boundary of a ball in a 2-dimensional, euclidean metric space?

An equivalence class of Gaussian numbers based on their norms?

Look at the beauty of the sin function and tell me that has more to do with triangles than the circle. sin and cos are the length of the projected vector along each axis as a point moves about the circle. connecting those points forms a triangle, sure, but it seems less fundamental.

I'd absolutely concede that the sine function is more about a circle than it is about a triangle. If you could really say that the sine function is about a geometry at all.