Hyperbolic Function Calculator
The hyperbolic functions sinh, cosh and tanh are built directly from the exponential function and are the hyperbola counterparts of the familiar circular trigonometric functions. They describe the shape of a hanging cable, appear in relativity, and serve as smooth activation functions in machine learning. Enter any real number x and this calculator returns the hyperbolic sine, hyperbolic cosine and hyperbolic tangent at once, each computed from the exact exponential definitions shown below. All three are defined for every real input.
Hyperbolic function definitions
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
Identity: cosh(x)^2 - sinh(x)^2 = 1
Each function is evaluated from the exponential. The identity output should always equal 1, providing a built-in check on the arithmetic.
About hyperbolic functions
- sinh is odd and cosh is even, mirroring sine and cosine.
- cosh is never below 1, since it is a sum of positive exponentials over two.
- tanh is bounded strictly between minus 1 and 1.
- A hanging cable hangs in a catenary, the graph of a scaled cosh.
- Hyperbolic functions appear in special relativity through the rapidity variable.
Hyperbolic functions: frequently asked questions
What are the hyperbolic functions?
The hyperbolic sine, cosine and tangent are defined from the exponential function: sinh(x) = (e^x - e^-x) / 2, cosh(x) = (e^x + e^-x) / 2, and tanh(x) = sinh(x) / cosh(x). They are the hyperbola analogues of the circular trig functions.
How do hyperbolic functions differ from trig functions?
Circular functions parametrise a circle, while hyperbolic functions parametrise a hyperbola. They obey cosh squared minus sinh squared equals 1, compared with cos squared plus sin squared equals 1 for the circular case.
What is the range of tanh?
tanh always lies strictly between -1 and 1. It is 0 at the origin, rises toward 1 for large positive x and falls toward minus 1 for large negative x, which is why it is popular as a smooth activation function.
Where is cosh seen in the real world?
A hanging chain or cable under its own weight forms a catenary, whose shape is exactly a scaled cosh curve. The Gateway Arch in St. Louis follows an inverted catenary.
Is sinh an odd or even function?
sinh is odd, so sinh(-x) equals minus sinh(x). cosh is even, so cosh(-x) equals cosh(x). tanh, being sinh over cosh, is odd.
Official sources
- NIST Digital Library of Mathematical Functions: Hyperbolic Functions: Definitions.
- NIST Digital Library of Mathematical Functions: Hyperbolic Functions: Identities.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.