Libraries compute sin(x) from a polynomial rather than from any geometric definition. Why is a polynomial the right thing for a processor to evaluate?

C. Because a CPU has only addition and multiplication to work with.

A processor adds and multiplies, so evaluating sine means reducing it to operations that exist, and a polynomial is exactly that. The polynomial is not equal to sine; it agrees near the expansion point, so implementations first fold the argument into a small range using periodicity. Nor are the coefficients Taylor's: minimax polynomials hit the same accuracy over an interval with fewer terms.