**Define Hyperbolic Function** in Simple Way

- Sin Hyperbolic Function
- Cos Hyperbolic Function

### Simply Define sinh x hyperbolic function

**Sinh x =** \frac12\left(e^x-\;e^{-x}\right) is called **hyperbolic sine** function. Its domain and range are the set of all real numbers.

**Formula for Cosine hyperbolic function in Math with Detail**

C**osh x =** \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

**Four Hyperbolic Functions and Formulas and Their Details**

The remaining four hyperbolic functions are defined in the terms of the hyperbolic **sine** and the hyperbolic **cosine** function as follows:

**Four Hyperbolic Function **Formulas are

Functions | is equal to | Formulas |

tanh x | \frac{\sin h\;x}{\cos h\;x} | \frac{e^x\;-\;e^{-x}}{e^x\;+\;e^{-x}} |

sech x | \frac1{\cos h\;x} | \frac2{e^x\;+\;e^{-x}} |

coth x | \frac{\cos h\;x}{\sin h\;x} | \frac{e^x\;+\;e^{-x}}{e^x\;+\;e^{-x}} |

csch x | \frac1{\sin h\;x} | \frac2{e^x\;-\;e^{-x}} |

*The hyperbolic functions have the same properties that resemble to those of trigonometric functions.*