# 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

\(\ds e^{-i z}\) | \(=\) | \(\ds \map \cos {-z} + i \map \sin {-z}\) | Euler's Formula | |||||||||||

\(\ds \) | \(=\) | \(\ds \cos z + i \map \sin {-z}\) | Cosine Function is Even | |||||||||||

\(\ds \) | \(=\) | \(\ds \cos z - i \sin z\) | Sine Function is Odd |

$\blacksquare$

## Also known as

**Euler's Formula** in this and its **corollary form**, along with **Euler's Sine Identity** and **Euler's Cosine Identity** are also found referred to as **Euler's Identities**.

However, this allows for confusion with Euler's Identity:

- $e^{i \pi} + 1 = 0$

It is wise when referring to it by name, therefore, to ensure that the equation itself is also specified.

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $\text{(ii)}$: $(4.16)$