But it's trigonometry, not anometry.

Really, the unit circle is important because it's the unit circle, not because it's the unit circle. A circle is 2π angles. A triangle is composed of the same kind of angles, which means that, because the diameter of the circle doesn't matter (because it's a unit circle), then it's less important what the lengths are than it is how the lengths relate to one another. I.e., what factor of a unit vector as compared to what other factor of a unit vector.

What's a circle if not all the right triangles (up to a choice of units)?

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.