# csch() Function & Examples

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

## csch

#### csch(x)

Try it yourself:

```calculio
csch(2)
1 / sinh(2)
```

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

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

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

---

## Calculate hyperbolic cosecant with csch

**csch(x)** returns the hyperbolic cosecant, the reciprocal of hyperbolic sine: `1 / sinh(x)`. It is useful in identities and solutions involving hyperbolic functions, exponential forms, and differential equations.

## Key behavior

For real x, csch is an odd function: changing the sign of x changes the sign of the result. Its magnitude becomes small as |x| grows, but it is undefined at zero because [sinh](https://calcul.io/function/sinh/index.md) is zero there. Compare it with [sech](https://calcul.io/function/sech/index.md), the reciprocal of cosh, which is defined at zero.

## Identity and example

At x = 1, `csch(1)` is the reciprocal of `sinh(1)`. A useful identity is `coth(x)^2 - csch(x)^2 = 1`; see [coth](https://calcul.io/function/coth/index.md) for the other term. The inverse function [acsch](https://calcul.io/function/acsch/index.md) can recover a value subject to its principal branch.

## Numerical cautions

Do not evaluate csch at zero, and treat values close to zero carefully because the reciprocal can be very large. For large magnitudes, exponential underflow or loss of precision may matter in an implementation. Use [exp](https://calcul.io/function/exp/index.md) when an explicitly scaled exponential expression is more stable for your model.

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