think think think

My university maths lecturer would have said: ‘are you sure you really want to do all that work?’. This of course applies to the solution of the taylor polynomial for cos. Why not drop an extra term, and use the identity cos(x) = cos(pi – x). So that’s exactly what I plan to try.
I figure I’ll get 25 times less error, and I can just use the standard quadratic formula to solve the equation which means it will be faster too. The intelligent part is knowing when to swap, but that’s easy because if your arc is longer than the chord by (C * pi / 2 – C) / 2 then the angle must be greater than 90.
The error is 25 times smaller at 90 than at 180 for the 1 term longed expansion, and I’m only really concerned with being accurate at 0 and 180, so a tiny error at 90 won’t bother me.

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