I'm being haunted by this problem! My only solution at this point is the dirtiest of programmers tricks(the lookup table), which is workable for the level of precision I need. However, I can't help but think there's a better solution.

a = sin(f(x) * n)
or
a = (exp(complex(0, f(x) * n)) - exp(-complex(0, f(x) * n))) / 2j

a and x are known

Thanks in advance for any help you might offer.