If you're being technical, it's usually not a Taylor series, it's a minimax series. (The difference is that Taylor series minimize error at a given value, whereas minimax is trying to minimize maximum error in a range).

In most general math library implementations (e.g., the library in glibc, musl, etc.), the implementation of sin, as with most functions, is going to be a polynomial evaluation. See, e.g., https://github.com/kraj/musl/blob/kraj/master/src/math/__cos... for the implementation in musl, or https://github.com/bminor/glibc/blob/master/sysdeps/ieee754/... for glibc's implementation.

Of course, if you're not using a standard math library implementation, you're probably preferring speed over accuracy, and so you might use a lookup table and linear interpolation to get a very coarse approximation instead.