> It's possible to do quite a lot without infinities in calculus and trigonometry
This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points."
Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them.
You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating.
https://en.m.wikipedia.org/wiki/Rational_trigonometry