# Cis

Cis notation is a polar notation for complex numbers. For all complex numbers $z$, we can write $z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta$. Notice that $\mathrm{cis}$ is made up by the first letter of $\cos$, $i$, and the first letter of $\sin$.

Once one gets used to the notation, it is almost always preferred to write $re^{i\theta}$ rather than $r\mathrm{cis }(\theta)$, as Euler's formula states that

$e^{i\theta} = \cos \theta + i \sin \theta.$

This is so that one can more naturally use the properties of the complex exponential. One important example is De Moivre's Theorem, which states that

$\mathrm{cis}(r\theta) = (\mathrm{cis}(\theta))^r.$

This is more easily understood in the complex exponential form:

$e^{i(r\theta)} = (e^{i\theta})^r.$