Inverse Hyperbolic Functions Calculator

Q: What is arsinh 2?

ln(2 + √5) ≈ 1.4436355, and sinh of that returns 2.

Q: Why is arcosh restricted to x ≥ 1?

Because cosh never dips below 1, so no real input produces a smaller output.

Work out inverse hyperbolic functions instantly with clear inputs, formula shown and shareable results.

About Inverse Hyperbolic Functions Calculator

Each inverse hyperbolic function has a closed logarithmic form: arsinh x = ln(x + √(x² + 1)) for all x, arcosh x = ln(x + √(x² − 1)) for x ≥ 1, and artanh x = ½ln((1 + x)/(1 − x)) for |x| < 1.

Formulas

arsinh

arsinh x = ln(x + √(x² + 1))

arcosh

arcosh x = ln(x + √(x² − 1)), x ≥ 1

artanh

artanh x = ½·ln((1 + x) / (1 − x)), |x| < 1

Worked example

With the default values below, Inverse Hyperbolic Functions Calculator returns Inverse hyperbolic value of 1.44363548. Change any input above and every figure here is recalculated in your browser.

Inputs used

Functionone of arsinh x (all x), arcosh x (x ≥ 1), artanh x (|x| < 1): arsinh x (all x)
Value of x: 2

Results

Inverse hyperbolic value: 1.44363548
Logarithmic form used: ln(x + √(x² + 1))
Applying the forward function back: 2.00000000
e raised to the result: 4.23606798

Step by step

1Values used
2arsinh
3arcosh
4artanh
5Inverse hyperbolic value= 1.44363548
6Logarithmic form used= ln(x + √(x² + 1))
7Applying the forward function back= 2.00000000
8e raised to the result= 4.23606798

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is arsinh 2?