Hyperbolic Functions Calculator

Background.

Hang a chain between two hooks and it sags into a curve that looks like a parabola but is not one. Its shape is y = a·cosh(x/a), the hyperbolic cosine, and the same small family of functions turns up wherever exponential growth and exponential decay are combined: a skydiver closing in on terminal velocity, the speed of ocean waves over a shallow seabed, the way velocities add in special relativity.

All three functions here are built from e^x and e^−x. The hyperbolic sine is half their difference, the hyperbolic cosine half their sum, and the hyperbolic tangent is the first divided by the second. They borrow their names from the circular functions for a geometric reason. The point (cos t, sin t) traces the unit circle x² + y² = 1, while the point (cosh t, sinh t) traces the right-hand branch of the unit hyperbola x² − y² = 1. So instead of sin² + cos² = 1 there is cosh²x − sinh²x = 1, the same identity with a minus sign.

Their behaviour is nothing like sine and cosine, though. Nothing repeats. sinh climbs steadily from large negative values to large positive ones, cosh never falls below 1 and is symmetric about zero, and tanh is squeezed between −1 and 1, which is why it serves as an activation function in neural networks. For large x, sinh and cosh both grow like e^x ÷ 2; at the page's upper limit of x = 100 they reach about 1.344 × 10⁴³.

What is hyperbolic functions calculator?

The hyperbolic functions are sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2 and tanh x = sinh x / cosh x, defined for every real x. sinh and tanh are odd functions, so flipping the sign of x flips the sign of the result; cosh is even, so it does not. sinh takes every real value, cosh takes values from 1 upward, and tanh stays strictly between −1 and 1. The argument x is a plain number, not an angle in degrees.

How to use this calculator.

1. 01Enter x, any real number from −100 to 100. If your formula has an argument such as x/a or k·h, work out that ratio first and enter the result.
2. 02Do not convert from degrees. The input is a dimensionless hyperbolic angle, so there is no degree or radian setting to choose.
3. 03Read sinh(x), cosh(x) and tanh(x) together; the first is the headline result, the other two appear alongside it.
4. 04To check hand calculations, confirm that cosh² − sinh² comes to 1 and that tanh equals sinh divided by cosh.

The formula.

sinh x = (e^x − e^−x) ÷ 2, cosh x = (e^x + e^−x) ÷ 2, and tanh x = (e^x − e^−x) ÷ (e^x + e^−x), where e ≈ 2.71828 is the base of natural logarithms. Because e^−x shrinks toward zero as x grows, sinh and cosh converge on the same value, e^x ÷ 2, and tanh approaches 1; for negative x the roles of the two exponentials swap. Calculus explains why the functions keep appearing in physics: the derivative of sinh is cosh and the derivative of cosh is sinh, with no sign change, so both satisfy y″ = y, much as sine and cosine satisfy y″ = −y. The derivative of tanh is 1 − tanh². The values here are worked out in high-precision decimal arithmetic and shown to 10 decimal places.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.