Based on my experience writing many games that work great barring the occasional random physics engine explosion, I suspect that trigonometry is responsible for a significant proportion of glitches.

I think over the years I subconsciously learned to avoid trig because of the issues mentioned, but I do still fall back to angles, especially for things like camera rotation. I am curious how far the OP goes with this crusade in their production code.

Your response is well-grounded--trig is trouble. Angles are often fine, but many 3rd party library functions are not.

Have you ended up with a set of self-implemented tools that you reuse?