Cube root of complex number without trigonometric functions
Solution 1:
There is no such nice formula for the cube root of a complex number with both real and imaginary parts nonzero. If you write out the real and imaginary parts of your cube root, you wind up solving cubic equations in one variable that have three irrational roots. This is the Casus Irreducibilis http://en.wikipedia.org/wiki/Casus_irreducibilis
In turn, for each of those cubics, Cardano's method leads you to finding the cube roots of other complex numbers. All very circular, and never gets anywhere.