I don't have a super-wide gamut of experience here and numerical analysis isn't my specialty, but nearly all trig implementations I've looked into (in both software and hardware) make heavy use of lookup tables and other shortcuts. I've never seen a Taylor series used in a general implementation - not saying it doesn't exist anywhere, but in most cases that I'm familiar with you could support turns just as easily with a different lookup table.

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.