This has been some sort of a mix of peeve and a moment of enlightenment of mine when I understood this.

I wholeheartedly agree with the point being made in the post. I had commented about this in the recent asin() post but deleted thinking it might not be of general interest.

If you care about angles and rotations in the plane, it is often profitable to represent an angle not by a scalar such as a degree or a radian but as a tuple

    (cos \theta, sin \theta)

or as a complex number.

This way one can often avoid calls to expensive trigonometric functions. One may need calls to square roots and general polynomial root finding.

In Python you can represent an angle as a unit complex numbers and the runtime will do the computations for you.

For example, if you needed the angular bisector of an angle subtended at the origin (you can translate the vertex there and later undo the translation), the bisector is just the geometric mean of the arms of the angle

   sqrt(z1 * z2)

Along with stereographic transform and its inverse you can do a lot.

This is directly related to the field of algebraic numbers.

With complex numbers you get translations, scaled rotations and reflections. Sufficient for Euclidean geometry.