# cosh() Function & Examples

Use the cosh() trigonometry function in Calcul.io. Review its syntax, edit working examples, understand the result, and explore related math functions.

## cosh

#### cosh(x)

Try it yourself:

```calculio
cosh(0.5)
```

[sinh](https://calcul.io/function/sinh/index.md)

[tanh](https://calcul.io/function/tanh/index.md)

[coth](https://calcul.io/function/coth/index.md)

---

## Work with hyperbolic cosine using cosh

**cosh(x)** returns the hyperbolic cosine, defined by `(e^x + e^-x) / 2`. Unlike ordinary cosine, it is not periodic. For real x it is always at least 1 and grows rapidly as the magnitude of x increases.

## Where cosh appears

Hyperbolic functions model shapes such as a hanging cable, as well as solutions to differential equations and transfer problems. A catenary can be expressed with a scaled and shifted `cosh`. Its companion [sinh](https://calcul.io/function/sinh/index.md) measures the odd hyperbolic component, and [tanh](https://calcul.io/function/tanh/index.md) is their ratio.

## Identity and example

The fundamental identity is `cosh(x)^2 - sinh(x)^2 = 1`. For example, evaluate `cosh(0)` to get 1, then compare the identity at a small nonzero value. The inverse [acosh](https://calcul.io/function/acosh/index.md) recovers a real x from cosh(x) when the input is at least 1.

## Input considerations

Real and complex values are supported, but large real magnitudes can overflow because exponentials grow quickly. Use [exp](https://calcul.io/function/exp/index.md) when you specifically need the exponential components, and use a scaled formulation in numerical work when values are extreme. Do not substitute [cos](https://calcul.io/function/cos/index.md) for cosh: although their names are similar, their graphs and identities are different.

[All functions](https://calcul.io/functions/index.md)