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Inverse Hyperbolic Functions — Definition, Formula & Examples

Inverse hyperbolic functions reverse the hyperbolic functions (sinh, cosh, tanh, etc.), taking an output of a hyperbolic function and returning the original input. Each one can be written as an explicit formula involving natural logarithms.

For a hyperbolic function hh, its inverse h−1h^{-1} satisfies h−1(h(x))=xh^{-1}(h(x)) = x on the appropriate domain. Because hyperbolic functions are defined in terms of exe^x, each inverse can be expressed in closed form using ln⁡\ln. For example, sinh⁡−1(x)=ln⁡ ⁣(x+x2+1)\sinh^{-1}(x) = \ln\!\left(x + \sqrt{x^2 + 1}\right) for all real xx.

Key Formula

sinh⁡−1(x)=ln⁡ ⁣(x+x2+1),cosh⁡−1(x)=ln⁡ ⁣(x+x2−1),tanh⁡−1(x)=12ln⁡ ⁣(1+x1−x)\sinh^{-1}(x) = \ln\!\left(x + \sqrt{x^2 + 1}\right), \quad \cosh^{-1}(x) = \ln\!\left(x + \sqrt{x^2 - 1}\right), \quad \tanh^{-1}(x) = \frac{1}{2}\ln\!\left(\frac{1+x}{1-x}\right)
Where:
  • xx = The input value; domain restrictions: all reals for sinh⁻¹, x ≥ 1 for cosh⁻¹, |x| < 1 for tanh⁻¹
  • ln⁡\ln = The natural logarithm (base e)

How It Works

To evaluate an inverse hyperbolic function, you can use its logarithmic formula. Set y=sinh⁡(x)=ex−e−x2y = \sinh(x) = \frac{e^x - e^{-x}}{2}, solve for xx in terms of yy using algebra (multiply through by exe^x, form a quadratic in exe^x, apply the quadratic formula), and you obtain the ln⁡\ln-based expression. The same strategy works for cosh⁡−1\cosh^{-1} and tanh⁡−1\tanh^{-1}. These functions appear frequently when integrating expressions like 1x2+1\frac{1}{\sqrt{x^2+1}}, where the antiderivative is sinh⁡−1(x)+C\sinh^{-1}(x) + C.

Worked Example

Problem: Evaluate sinh⁻¹(0) using the logarithmic formula.
Write the formula: Apply the closed-form expression for inverse sinh.
sinh⁡−1(0)=ln⁡ ⁣(0+02+1)\sinh^{-1}(0) = \ln\!\left(0 + \sqrt{0^2 + 1}\right)
Simplify inside the logarithm: Compute the square root and add.
=ln⁡ ⁣(1)=ln⁡(1)= \ln\!\left(\sqrt{1}\right) = \ln(1)
Evaluate: The natural log of 1 is 0.
=0= 0
Answer: sinh⁡−1(0)=0\sinh^{-1}(0) = 0

Why It Matters

Inverse hyperbolic functions show up as antiderivatives of common algebraic expressions in Calculus II, such as ∫dxx2+a2\int \frac{dx}{\sqrt{x^2 + a^2}}. They also appear in physics — for instance, the shape of a hanging cable (catenary) involves cosh⁡\cosh, and solving for arc length or position along it requires cosh⁡−1\cosh^{-1}.

Common Mistakes

Mistake: Confusing cosh⁡−1(x)\cosh^{-1}(x) domain with sinh⁡−1(x)\sinh^{-1}(x) domain.
Correction: sinh⁡−1(x)\sinh^{-1}(x) accepts all real xx, but cosh⁡−1(x)\cosh^{-1}(x) requires x≥1x \geq 1 because cosh⁡\cosh outputs values in [1,∞)[1, \infty).