Hyperbolic trigonometric functions

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The hyperbolic trigonometric functions, also referred to as simply "hyperbolic functions," are analogous to the standard trigonometric functions using a hyperbola as the defining conic section rather than a circle.[1] This has the effect of removing any i's that appear in the complex definition of the standard trigonometric functions. As such, they tend to differentiate in an analogous way to standard trigonometric functions, up to perhaps a negative sign. They are defined as:

  • Hyperbolic sine:
  • Hyperbolic cosine:
  • Hyperbolic tangent:
  • Hyperbolic cosecant:
  • Hyperbolic secant:
  • Hyperbolic cotangent:


Sinh-cosh.png Tanh-coth.png
sinh and cosh tanh and coth


The hyperbolic trigonometric functions have many identities that are similar to those of trigonometric functions. These can be remembered by replacing instances of with . For example, the trigonometric identity,

has the corresponding hyperbolic identity,

Other identities include:

Also, note that:


  1. Hyperbolic functions from mathworld.wolfram.com

See also