Inverse Hyperbolic Sine (arcsinh) Calculator

This calculator works out the inverse hyperbolic sine of any number, written as arcsinh(x) or asinh(x), which is the value whose hyperbolic sine equals x. Rather than solving sinh(y) = x by trial and error, the calculator applies the closed logarithmic form arcsinh(x) = ln(x + the square root of (x squared plus one)) and returns the exact answer instantly. You only need to enter one figure, the value of x, into the input field, and there are no restrictions on what you can type: hyperbolic sine covers every real number, so its inverse accepts any real input, positive, negative, whole or decimal, and always returns a single real result. Alongside the main answer, the calculator shows a built-in check by feeding the result back through the hyperbolic sine function, so you can confirm that sinh of the answer returns your original input x, giving you confidence the working is correct. This function turns up in calculus when integrating expressions involving square roots, in physics problems and engineering models, and in statistics, where it is used as a variance stabilising transform for skewed data that can include zero or negative values, unlike a log transform. Simply type in your value of x and the result updates automatically as you type. A full definition, a worked example and the underlying assumptions are set out below the calculator.

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Value of x

0.881374

Check: sinh(result)1

Input x1

The formula

The inverse hyperbolic sine has a closed form in logarithms: arcsinh(x) = ln(x + sqrt(x^2 + 1)). It is defined for every real x and is an odd function.

Worked example

arcsinh(1) = ln(1 + sqrt(2)), about 0.881374. The hyperbolic sine of 0.881374 returns 1. Enter 1 above to confirm.

Frequently asked questions

What is the formula for arcsinh?

arcsinh(x) = ln(x + sqrt(x^2 + 1)), a closed form using the natural logarithm.

What inputs are allowed?

Any real number. Hyperbolic sine covers all real values, so its inverse has no domain restriction.

Is arcsinh the same as 1 / sinh?

No. arcsinh is the inverse function of sinh, not its reciprocal. The reciprocal of sinh is the hyperbolic cosecant.

Who this calculator is for

This calculator is for students, engineers and anyone needing the inverse hyperbolic sine.

What this calculator assumes

You enter a real number x.
arcsinh is computed from its logarithmic form.
Results are rounded for display.

Formula and sources

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