nLab Euler's formula

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 $\sin(x) \coloneqq \frac1{2 i}(\exp(i x) - \exp(-i x))$).

Related concepts

• trigonometric identity

References

Named after Leonhard Euler.

• Wikipedia, Euler’s formula