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# Contents

Euler’s formula is a fundamental relation between the exponential function and the trigonometric functions.

## Statement

The exponential function $\exp: \mathbb{C} \to \mathbb{C}$ relates to sine and cosine $\cos, \sin \colon \mathbb{R} \to \mathbb{R}$ via

$\exp(i x) = \cos(x) + i \sin(x)$

This can be interpreted as either a theorem (especially when $x$ is restricted to real numbers and $\cos, \sin$ have been defined independently of $\exp$), or as an implicit definition of $\cos, \sin$ (where more explicitly we have $\cos(x) \coloneqq \frac1{2}(\exp(i x) + \exp(-i x))$ and $\sin(x) \coloneqq \frac1{2 i}(\exp(i x) - \exp(-i x))$).

## References

Named after Leonhard Euler.

Last revised on May 10, 2019 at 11:09:15. See the history of this page for a list of all contributions to it.