---
tag: 0J6M03WSS6
kind: definition
name: Hyperbolic functions
msc: [33B10]
defines: 01PPDSDP2N
uses: [0GWD21EWRW, 0EQFWKJ37J, 0MWYQTZ07Z]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_functions
    locator: Exponential definitions
---

## Definition

The hyperbolic functions are the functions $\sinh$, $\cosh$ and $\tanh$ on $\mathbb{R}$, together with their reciprocals:
$$\coth x = \frac{\cosh x}{\sinh x} \ (x \ne 0), \qquad \operatorname{sech} x = \frac{1}{\cosh x}, \qquad \operatorname{csch} x = \frac{1}{\sinh x} \ (x \ne 0).$$
Here [[$\sinh$|0GWD21EWRW]], [[$\cosh$|0EQFWKJ37J]] and [[$\tanh$|0MWYQTZ07Z]] are the hyperbolic sine, cosine and tangent.

## Remarks

They satisfy identities similar to the trigonometric identities, often with a change of sign.
The point $(\cosh t, \sinh t)$ lies on the hyperbola $x^2 - y^2 = 1$, as $(\cos t, \sin t)$ lies on the circle.
Vincenzo Riccati introduced the hyperbolic functions in 1757.

## Explanations

### L1 human

The hyperbolic functions are built from $e^x$ and $e^{-x}$.
They play the role for the hyperbola that sine and cosine play for the circle.