# acosh() Function & Examples

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

## acosh

#### acosh(x)

Try it yourself:

```calculio
acosh(1.5)
```

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

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

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

---

## acosh(): The Inverse Hyperbolic Cosine

**acosh()** returns the inverse hyperbolic cosine of a value. It undoes [cosh()](https://calcul.io/function/cosh/index.md) on its principal real branch, making it useful in hyperbolic geometry, catenary curves, and formulas involving exponential growth. Unlike circular cosine, hyperbolic cosine is built from exponentials rather than rotation around a circle.

## Valid real inputs

For real arithmetic, acosh(x) is defined for x greater than or equal to 1 and returns a non-negative value. Thus acosh(1) is 0. Values below 1 have no real result, although they can be evaluated in complex arithmetic. Check constraints early, especially when x is computed from measured data.

One equivalent form is acosh(x) = log(x + sqrt(x² − 1)). This relation explains why values near 1 require care: subtraction and square roots can magnify rounding effects. Use [sqrt()](https://calcul.io/function/sqrt/index.md) and [log()](https://calcul.io/function/log/index.md) for an explicit formula when you need to inspect intermediate values, but prefer acosh for a direct, readable expression.

## Practical use

Hyperbolic functions model hanging cables, relativistic transformations, and solutions to differential equations. To verify a result, apply cosh(acosh(x)); within numerical precision it returns x. Related functions include [asinh()](https://calcul.io/function/asinh/index.md) and [atanh()](https://calcul.io/function/atanh/index.md), whose domains differ significantly.

Do not confuse acosh with acos. Their similar names refer to different families: acos works with circular angles and accepts real values from −1 to 1, while acosh works with hyperbolic values and accepts real values starting at 1.

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