Euler's Identities
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Theorem
The following identities are referred to as Euler's Identities:
Euler's Formula
Let $z \in \C$ be a complex number.
Then:
- $e^{i z} = \cos z + i \sin z$
Euler's Sine Identity
- $\sin z = \dfrac {e^{i z} - e^{-i z} } {2 i}$
Euler's Cosine Identity
- $\cos z = \dfrac {e^{i z} + e^{-i z} } 2$
On $\mathsf{Pr} \infty \mathsf{fWiki}$, the following are also classified as members of the set of Euler's Identities:
Euler's Tangent Identity
- $\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$
Euler's Cotangent Identity
- $\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$
Euler's Secant Identity
- $\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$
Euler's Cosecant Identity
- $\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$
where:
- $z \in \C$ is a complex number
- $i$ denotes the imaginary unit: $i^2 = -1$
Source of Name
This entry was named for Leonhard Paul Euler.
Historical Note
Euler's Identities were documented by Leonhard Paul Euler some time around $1748$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Euler's identities
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Euler's identities