What are inverse hyperbolic functions?

Hyperbolic functions \(\sinh, \cosh, \tanh, \sech, \csch, \coth\) are analogues of the familiar trigonometric functions, but based on hyperbolas instead of circles. Their inverse functions undo the action of the hyperbolic functions:

• \(y = \operatorname{asinh}(x)\) means \(x = \sinh(y)\).
• \(y = \operatorname{acosh}(x)\) means \(x = \cosh(y)\).
• \(y = \operatorname{atanh}(x)\) means \(x = \tanh(y)\).
• \(y = \operatorname{asech}(x)\) means \(x = \sech(y)\).
• \(y = \operatorname{acsch}(x)\) means \(x = \csch(y)\).
• \(y = \operatorname{acoth}(x)\) means \(x = \coth(y)\).

They arise naturally in integration, in solutions of certain differential equations, in special relativity, and in various engineering models where hyperbolic functions appear.

Logarithmic formulas for inverse hyperbolic functions

Each inverse hyperbolic function can be written using only square roots and natural logarithms:

Our calculator uses these identities (or equivalent stable forms) internally and shows them in the step-by-step explanation so you can follow the derivation.

Domain and range

For real-valued inverse hyperbolic functions, the domains are:

• \(\operatorname{asinh}(x)\): all real \(x\), range all real values.
• \(\operatorname{acosh}(x)\): \(x \ge 1\), range \([0, +\infty)\).
• \(\operatorname{atanh}(x)\): \(|x| < 1\), range all real values.
• \(\operatorname{asech}(x)\): \(0 < x \le 1\), range \([0, +\infty)\).
• \(\operatorname{acsch}(x)\): all real \(x \ne 0\), range all real values.
• \(\operatorname{acoth}(x)\): \(|x| > 1\), range all real values.

The calculator validates these conditions before computing. If you are working in complex analysis, you will need a more advanced tool that supports complex values and branch cuts.

Examples

• \(\operatorname{asinh}(1.5) = \ln\bigl(1.5 + \sqrt{1.5^2 + 1}\bigr)\).
• \(\operatorname{acosh}(2) = \ln\bigl(2 + \sqrt{2-1}\sqrt{2+1}\bigr) = \ln(2 + \sqrt{3})\).
• \(\operatorname{atanh}(0.5) = \tfrac12 \ln\!\left(\frac{1.5}{0.5}\right) = \tfrac12 \ln(3)\).
• \(\operatorname{asech}(0.8) = \ln\!\left(\frac{1 + \sqrt{1 - 0.8^2}}{0.8}\right).\)

FAQ: using the inverse hyperbolic functions calculator

Why do my inputs sometimes give a domain error?

Unlike asinh(x), which accepts any real x, acosh(x), atanh(x), asech(x), acsch(x), and acoth(x) all have restricted domains in the real numbers. For example, acosh(x) only makes sense for \(x \ge 1\), and atanh(x) only for \(|x| < 1\). The calculator blocks invalid inputs instead of returning a meaningless number.

Do inverse hyperbolic functions use degrees or radians?

Hyperbolic functions and their inverses are typically treated as mappings on the real line, not on a circle, so the output is dimensionless rather than expressed in “degrees”. When you see these functions in calculus or physics, the values are implicitly in the natural (radian-like) scale used by the exponential function.

How is this different from inverse trigonometric functions?

Inverse trigonometric functions (arcsin, arccos, arctan) come from sine and cosine of angles on the unit circle, while inverse hyperbolic functions come from hyperbolas and exponential combinations like \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). Their algebraic identities, domains, and applications are different, even though the names sound similar.

Can I trust this tool for research or engineering reports?

The calculator uses standard double-precision floating-point arithmetic, comparable to most scientific calculators and programming languages. It is appropriate for learning and many practical tasks, but for mission-critical work you should:

• Validate the formulas and units against your discipline’s standards or textbooks.
• Check that the numerical precision is sufficient for your error budget.
• Replicate key results using independent software when possible.