There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance.

Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function.

That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway.

Still, I can dream!

[0] https://frinklang.org/

[1] https://en.wikipedia.org/wiki/Exact_trigonometric_values