Navigation with a sextant or maneuvers using gimbal angles might be two examples of cases where an apollo computer might need to do trigonometry.

sine, cosine, tangent, etc. are transcendental functions, as are logarithms. You can't evaluate these functions with a simple expression built on multiplication or division for example, without at least an iterative algorithm.

An engineer on the ground would grab a slide rule for two or three digits, and for more go to a book of trig, log and other tables for more digits. Between two lines you could interpolate by hand for even more digits.

But how did the apollo computers evaluate transcendental functions, or how at least were calculations that required the use of transcendental functions implemented in the programs?