How do GPUs compute sines?
I believe that NVidia GPUs they use a table lookup, followed by a quadratic interpolation.  I think they are using an algorithm similar to the one described in: Oberman, Stuart F; Siu, Michael Y: "A High-Performance Area-Efficienct Mutlifunction Interpolator," _IEEE Int'l Symp Comp Arithmetic, (ARITH-17):272-279, 2005.

The table lookup is indexed with the $m$ most-significant bits from the input, $x$, and returns three coefficients, $c_0$, $c_1$, $c_2$.  The final result is produced by evaluating $c_0 + c_1 x + c_2 x^2$.  The coefficients for each range of $x$ are chosen to minimize the maximum error from the target function over that range.

So that the unit can be fully pipelined to produce one result per cycle the unit contains a special squaring unit and two booth-encoded wallace-tree multipliers.  For each special function they choose the number of table entries ($2^m$) so that the polynomial evaluation will give them a single-precision IEEE FP answer correct to within a couple of units in the last place.