Euler's Identities

Theorem

The following identities are referred to as Euler's Identities:

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

Then:

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

$\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$

$\cos z = \dfrac {e^{i z} + e^{-i z} } 2$

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the following are also classified as members of the set of Euler's Identities:

$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$

$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$

$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$

$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$

where:

$z \in \C$ is a complex number

$i$ denotes the imaginary unit: $i^2 = -1$

This entry was named for Leonhard Paul Euler.

Euler's Identities were documented by Leonhard Paul Euler some time around $1748$.