I guess the point is: How often do we really need actual angles in the code? Probably only at the very ends: input from users and output to users. Everywhere else, we should just be treating them as sin/cos pairs or dot/cross pairs. So when the user inputs an angle, immediately convert it to what the computer actually needs, store it that way throughout the computation, and then only if/when the user needs to see an actual angle would you need to convert it back.

This is how most physics/graphics engines work.