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.