The sin(x) = x approximation is actually exact (in terms of doubles) for |x| < 2^-26 = 1.4e-8. See also [1]. This happens to cover 48.6% of all doubles.
Similarly, cos(x) = 1 for |x| < 2^-27 = 7.45e-9 (see [2]).
Finally, sin(double(pi)) tells you the error in double(pi) - that is, how far the double representation of pi is away from the "real", mathematical pi [3].