# Hyperbolic functions

input sqrt(2) for square root of 2 for example.

## Hyperbolic functions in R (real numbers)

Function | Abbreviation | Domain | Range |
---|---|---|---|

hyperbolic sine | y = sinh(x) | All real numbers | All real numbers |

hyperbolic cosine | y = cosh(x) | All real numbers | y >= 1 |

hyperbolic tangent | y = tanh(x) | All real numbers | -1 < y < 1 |

hyperbolic cotangent | y = coth (x) | all non-zero real numbers | y < -1 ou y > 1 |

## hyperbolic sine

The hyperbolic sinus function is defined as follows,

`sinh(x) = (e^x - e^ (-x)) /2`

sinh(x) is defined for all real numbers x so the definition domain is `RR`.

The range is `RR`.

## hyperbolic cosine

The hyperbolic cosine function is defined as follows,

`cosh (x) = (e^x + e^ (-x)) /2`

cosh(x) is defined for all real numbers x so the definition domain is `RR`.

The range (set of function values) is [1, +∞[.

## hyperbolic tangent

The hyperbolic tangent is defined as the ratio between the hyperbolic sine and the hyperbolic cosine functions.

`tanh (x) = frac {sinh (x)} {cosh (x)} = frac {e^x + e^ (-x)} {e^x + e^ (-x)} = frac {e^ (2x) - 1} {e^ (2x) + 1} `

tanh(x) is defined for all real numbers x so the definition domain is `RR`.

The range (set of function values) is] -1, 1 [.

## hyperbolic cotangent

The hyperbolic cotangent is defined as the ratio between the hyperbolic cosine and the hyperbolic sine functions.

`coth (x) = frac {cosh (x)} {sinh (x)} = frac {e^x - e^ (-x)} {e^x - e^ (-x)} = frac {e^ (2x) + 1} {e^ (2x) - 1} `

It can also defined as the hyperbolic tangent reciprocal,

`coth(x) = frac {1} {tanh (x)}`

coth(x) is defined for all non-zero real numbers so the definition domain is the set of nonzero real: `RR`\ {0}.

The range is ] -∞, -1 [ U ] 1, +∞ [.