# Hyperbolic Functions

## Exponential functions related to the hyperbola

### Hyperbolic Functions

Hyperbolic functions are combinations of the exponentials *e ^{x}* and

*e*.

^{-x}*sinh x = (e*^{x}* - e*^{-x}*) / 2 (1)*

*cosh x = (e*^{x}* + e*^{-x}*) / 2 (2)*

*tanh x = sinh x / cosh x = (e*^{x}* - e*^{-x}*) / (e*^{x}* + e*^{-x}*) (3)*

### Values for Hyperbolic Functions

x | sinh x | cosh x | tanh x |
---|---|---|---|

0 | 0.00000 | 1.0000 | 0.00000 |

0.1 | 0.10017 | 1.00500 | 0.09967 |

0.3 | 0.30452 | 1.04534 | 0.29131 |

0.4 | 0.41075 | 1.08107 | 0.37995 |

0.5 | 0.52110 | 1.12763 | 0.46212 |

0.6 | 0.63665 | 1.18547 | 0.53705 |

0.7 | 0.75858 | 1.25517 | 0.60437 |

0.8 | 0.88811 | 1.33743 | 0.66404 |

0.9 | 1.02652 | 1.43309 | 0.71630 |

1.0 | 1.17520 | 1.54308 | 0.76159 |

1.1 | 1.33565 | 1.66852 | 0.80050 |

1.2 | 1.50946 | 1.81066 | 0.83365 |

1.3 | 1.69838 | 1.97091 | 0.86172 |

1.4 | 1.90430 | 2.15090 | 0.88535 |

1.5 | 2.12928 | 2.35241 | 0.90515 |

1.6 | 2.37557 | 2.57746 | 0.92167 |

1.7 | 2.64563 | 2.82832 | 0.93541 |

1.8 | 2.94217 | 3.10747 | 0.94681 |

1.9 | 3.26816 | 3.41773 | 0.95624 |

2.0 | 3.62686 | 3.76220 | 0.96403 |

2.5 | 6.05020 | 6.13229 | 0.98661 |

3.0 | 10.0179 | 10.0677 | 0.99505 |

3.5 | 16.5426 | 16.5728 | 0.99818 |

4.0 | 27.2899 | 27.3082 | 0.99933 |

4.5 | 45.0030 | 45.0141 | 0.99975 |

5.0 | 74.2032 | 74.2099 | 0.99991 |

### Inverse Hyperbolic Functions

*sinh ^{-1} x = log_{e}(x + (x^{2} + 1)^{1/2})*

*(4)*

*cosh ^{-1} x = *

*log*x ≥ 1 (5)_{e}(x ± (x^{2}+ 1)^{1/2})*tanh ^{-1} x = 1/2 *

*(6)**log*x_{e}((1 + x) / (1 - x))^{2}‹ 1