# Defintion

## Definition of Hyperbolic Functions

• Hyperbolic functions are analogue to the circular trigonometric functions, but related to the hyperbola rather than the circle.
• The hyperbolic sine function, or sinh, is the function defined as `sinh(x) = 0.5(e^x - e^-x)`.
• The hyperbolic cosine function, or cosh, is the function defined as `cosh(x) = 0.5(e^x + e^-x)`.
• The hyperbolic tangent function, or tanh, is defined as the ratio of the hyperbolic sine to the hyperbolic cosine: `tanh(x) = sinh(x) / cosh(x)`.
• The hyperbolic functions are all functions of real variables.
• The hyperbolic functions often appear in solutions to differential equations and geometry problems involving hyperbolic shapes.
• The hyperbolic functions, much like the circular trigonometric functions, have similar identities that relate them to each other.

## Properties of Hyperbolic Functions

• The function `sinh(x)` is odd; that is, `sinh(-x) = -sinh(x)`.
• The function `cosh(x)` is even; that is, `cosh(-x) = cosh(x)`.
• The function `tanh(x)` is odd.
• For all real values of x, `cosh^2(x) - sinh^2(x) = 1`. This is similar to the Pythagorean identity for the circular functions, `cos^2(t) + sin^2(t) = 1`.
• The graphs of hyperbolic functions are hyperbolae.
• The functions `sinh(x)` and `cosh(x)` are inverses of each other.

## Calculating with Hyperbolic Functions

• Hyperbolic functions can be combined and manipulated just like any other functions to simplify or rearrange expressions.
• The derivatives and integrals of the hyperbolic functions can be found using the standard rules of calculus.
• Hyperbolic functions are particularly useful in integration due to the identities that enable integration by substitution and by parts.