# Hyperbolic Functions Calculator

URL: https://speedycalculator.com/calculators/math/hyperbolic-functions/
Last updated: 2026-10-30

> The hyperbolic functions calculator evaluates sinh, cosh, tanh, coth, sech and csch of any real number.

## Inputs

- x

## Results

- sinh(x)
- cosh(x)
- tanh(x)
- coth(x)
- sech(x)
- csch(x)

## 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
```

## Explanation

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:

xsinh(x)0.50.52110.750.822311.17521.251.60191.52.129323.6269

## 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.

## FAQ

**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)).


## Sources

- [OpenStax – Calculus Volume 1](https://openstax.org/details/books/calculus-volume-1)
- [Wolfram MathWorld – Hyperbolic functions](https://mathworld.wolfram.com/HyperbolicFunctions.html)