Hyperbolic functions are built from exponentials and play the same role for the hyperbola x² − y² = 1 that sine and cosine play for the circle. A hanging chain takes the shape of cosh (a catenary), tanh is a common activation function in neural networks, and they appear in relativity and heat transfer.
How to use the Hyperbolic Functions Calculator
- x
Results update as you type, and everything is calculated in your browser.
How it works
Each function is computed directly from eˣ and e⁻ˣ. coth and csch are undefined at x = 0.
What each result means
- sinh(x) = (exp([x]) – exp(-[x]))÷2
- cosh(x) = (exp([x]) + exp(-[x]))÷2
- tanh(x) = (exp([x]) – exp(-[x]))÷(exp([x]) + exp(-[x]))
- sech(x) = 2÷(exp([x]) + exp(-[x]))
Example values
How the main result changes when x changes and everything else stays at the example values above:
| x | sinh(x) |
|---|---|
| 0.5 | 0.5211 |
| 0.75 | 0.8223 |
| 1 | 1.1752 |
| 1.25 | 1.6019 |
| 1.5 | 2.1293 |
| 2 | 3.6269 |
Formula
sinh x = (eˣ − e⁻ˣ)/2 cosh x = (eˣ + e⁻ˣ)/2 tanh x = sinh x / cosh x
coth x = 1/tanh x sech x = 1/cosh x csch x = 1/sinh x
Worked example
Inputs — x: 1.
Results — sinh(x) = 1.1752011936; cosh(x) = 1.5430806348; tanh(x) = 0.761594156; coth(x) = 1.3130352855; sech(x) = 0.6480542737; csch(x) = 0.8509181282.
Key terms
- Catenary
- The curve y = a·cosh(x/a) formed by a hanging chain.
Frequently asked questions
What is cosh² − sinh²?
Always 1 – the hyperbolic version of sin² + cos² = 1.
Why are they called hyperbolic?
The point (cosh t, sinh t) lies on the hyperbola x² − y² = 1.
What is tanh used for?
It squashes numbers into (−1, 1), so it is used as an activation function in neural networks.
What are the inverse functions?
arsinh x = ln(x + √(x² + 1)), arcosh x = ln(x + √(x² − 1)), artanh x = ½ ln((1 + x)/(1 − x)).