Euler's Formula/Corollary

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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

$\blacksquare$


Sources