As with many subject that we learn early in school, it's often interesting revisiting them as adult to perceive additional layer of depth by casting a new look.

With trigonometry we tend to associate it with circle. But fundamentally it's the study of tri-angles.

What is interesting is that the whole theory is "relative". I would reference the wikipedia page for angle but it may make me look like an LLM. The triangle doesn't have positions and orientation baked-in, what matters is the length of the sides and the angle between them.

The theory by definition becomes translation and rotation invariant. And from this symmetry emerge the concept of rotations.

What is also interesting about the concept of angle is that it is a scalar whereas the original objects like lines live in an higher dimension. To avoid losing information you therefore need multiple of these scalars to fully describe the scene.

But there is a degree of redundancy because the angles of a triangle sums to pi. And from this degree of freedom results multiple paths to do the computations. But with this liberty comes the risks of not making progress and going in circles. Also it's harder to see if two points coming from different paths are the same or not, and that's why you have "identities".

Often for doing the computation it's useful to break the symmetry, by picking a center, even though all points could be centers, (but you pick one and that has made all the difference).

Similar situation arise in Elliptic Curve Cryptography, where all points could have the same role, but you pick one as your generator. Also in physics the concept of gauge invariance.