An Imaginary Tale: De Moivre’s Formula in a Gif

I’ve been reading the book, An Imaginary Tale: The Story of i [the square root of minus one] by Paul J. Nahin and it’s really interesting!  I’m almost to chapter 4 and so far it’s been introducing some nifty ideas about complex numbers.  Among the content, we find a very convenient way of calculating the value of a complex number when multiplied by itself an integer amount of times.  That is, $z^n=(a+bi)^n=r^n \cdot \left[ \cos\left(n\theta\right)+i\sin(n\theta)\right]\mid n\in \mathbb{Z}^+$.  This is De Moivre’s Formula.