[aidenn0]

Just using the 32 entry LUT and the small-angle approximations (sin(x) = x, cos(x) = 1-x^2/2) lets you calculate sin(x) within +/- 0.00015, which is rather impressive for something that can be done quickly by hand.

If you use cos(x) = 1 then you are still accurate to within 1%

[edit]

I think I also found an error in TFA? It seems like picking the "best" N would allow r to be in the range -pi/32 < r < pi/32, which makes the 3rd order Taylor series have an error of 4e-12, significantly better than the error range for 0 < r < pi/16