# Hyperbolic functions/R/Introduction/Section

## Definition

The function defined for ${\displaystyle {}x\in \mathbb {R} }$ by

${\displaystyle {}\sinh x:={\frac {1}{2}}{\left(e^{x}-e^{-x}\right)}\,,}$
is called hyperbolic sine.

## Definition

The function defined for ${\displaystyle {}x\in \mathbb {R} }$ by

${\displaystyle {}\cosh x:={\frac {1}{2}}{\left(e^{x}+e^{-x}\right)}\,,}$
is called hyperbolic cosine.

## Lemma

The functions hyperbolic sine and hyperbolic cosine

have the following properties.
1. ${\displaystyle {}\cosh x+\sinh x=e^{x}\,.}$
2. ${\displaystyle {}\cosh x-\sinh x=e^{-x}\,.}$
3. ${\displaystyle {}(\cosh x)^{2}-(\sinh x)^{2}=1\,.}$

### Proof

${\displaystyle \Box }$

## Lemma

The function hyperbolic sine is strictly increasing, and the function hyperbolic cosine is strictly decreasing on ${\displaystyle {}\mathbb {R} _{\leq 0}}$ and strictly increasing on ${\displaystyle {}\mathbb {R} _{\geq 0}}$.

### Proof

See exercise and exercise.

${\displaystyle \Box }$

## Definition

The function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \tanh x={\frac {\sinh x}{\cosh x}}={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}},}$
is called hyperbolic tangent.