They can't use the trig functions to get the trig functions. I guess the only way to evaluate complex exponents is with the Taylor's power series, which will only give you a decimal approximation depending on how far you want to go.
How can you find, say, sin(41.3°) without using a calculator? Or maybe a better question is: How does a calculator find that value when you punch it in? Also, what about arcsin, arccos, etc... How does a calculator find those values?