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):

- Why i? [http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stu...]

- An Introduction to Geometric Algebra with an Application in Rigid Body Mechanics [https://www.researchgate.net/profile/Terje-Vold/publication/...]

- Functions of Multivector Variables [https://journals.plos.org/plosone/article/file?id=10.1371/jo...]

- Lie Group Theory - A Completely Naive Introduction [https://jakobschwichtenberg.com/naive-introduction-lie-theor...]

- Previous HN discussion: Intuitive Understanding of Euler’s Formula [https://news.ycombinator.com/item?id=18325865]