# Euler's Formula/Corollary

## Corollary to Euler's Formula

Let $z \in \C$ be a complex number.

Then:

$e^{-i z} = \cos z - i \sin z$

where:

$e^{-i z}$ denotes the complex exponential function
$\cos z$ denotes the complex cosine function
$\sin z$ denotes complex sine function
$i$ denotes the imaginary unit.

### Corollary

This result is often presented and proved separately for arguments in the real domain:

$e^{-i \theta} = \cos \theta - i \sin \theta$

## Proof

 $\displaystyle e^{-i z}$ $=$ $\displaystyle \cos \paren {-z} + i \sin \paren {-z}$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \cos z + i \sin \paren {-z}$ Cosine Function is Even $\displaystyle$ $=$ $\displaystyle \cos z - i \sin z$ Sine Function is Odd

$\blacksquare$