What this calculator does
Hyperbolic functions are built from exponentials rather than circles, but they mirror the trigonometric functions closely enough that the notation is deliberately parallel. The key identity swaps a sign: cosh squared minus sinh squared equals 1, where the circular version adds.
They describe real shapes. A hanging chain or cable settles into a catenary, which is a cosh curve, and tanh appears throughout physics and machine learning as a smooth function bounded between minus one and one.
The formula
sinh is the difference of e to the x and e to the minus x over two, cosh is their sum over two, and tanh is the ratio of the two.
| Term | Meaning |
|---|---|
| sinh and cosh | Built from exponentials, mirroring sine and cosine. |
| Catenary | The curve of a hanging chain, which is a cosh curve. |
| Fundamental identity | cosh²x − sinh²x = 1, with a minus where the circular version has a plus. |
The inputs explained
| Field | What to enter |
|---|---|
| x | The value to evaluate at. Unlike trigonometric functions there is no angle involved, so any real number works. |
When to use it
Modelling a hanging cable
Suspension bridges and power lines follow cosh curves.
Neural network activations
tanh is a standard activation function, bounded between minus one and one.
Special relativity
Rapidity uses hyperbolic functions where velocity uses ordinary ratios.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do the three functions behave?
The same three functions across a range of x.
Questions
Why are they called hyperbolic?
Because the point (cosh t, sinh t) traces a hyperbola, exactly as (cos t, sin t) traces a circle. The parallel naming reflects that parallel construction rather than any direct relationship between the two families.
Why does the identity have a minus sign?
Because it comes from the hyperbola equation x² − y² = 1 rather than the circle equation x² + y² = 1. That single sign difference propagates through every hyperbolic identity.
Why is tanh bounded?
Because it is the ratio of sinh to cosh, and cosh always exceeds sinh in magnitude. As x grows both approach the same exponential, so their ratio approaches 1 without ever reaching it.
What is atanh restricted to?
Inputs strictly between minus one and one, since tanh only produces values in that range. Asking for atanh of 1 or beyond has no real answer, which the calculator reports rather than returning infinity.
For ordinary trigonometry, see the trigonometry calculator. For exponential equations, see the exponential equation calculator.