With complex numbers in polar form the tangent is easy. In the unitary circle, if you derive e^(i*t) you get i*e^(i*t), which maps the cos(t) real component to i*cos(t) imaginary, and the i*sin(t) imaginary component to -sin(t) real. This is effectively a 90 degree rotation, so if you integrate the tangent infinitesimally over its path parameterized by t you will recover the circle.
Here is some introductory material to what I referenced above and some generalizations into more dimensions (which, as Hamilton discovered when stumbling into quaternions trying to augment complex numbers, is not as straightforward as you would think):