What are hyperbolic functions?

Hyperbolic functions are analogues of trigonometric functions defined in terms of exponential functions. The basic ones are sinh, cosh, and tanh, defined via e^x as sinh x = (e^x − e^−x)/2, cosh x = (e^x + e^−x)/2, and tanh x = sinh x / cosh x. They naturally appear in differential equations, special relativity, and the shape of a hanging cable (catenary).

What does this hyperbolic functions calculator do?

The calculator computes the core hyperbolic functions sinh, cosh, tanh and their reciprocals coth, sech, and csch for a real argument x. You can enter x in radians or in degrees, choose the rounding precision, and see how closely the identity cosh²x − sinh²x = 1 is satisfied numerically.

Should I enter x in radians or degrees?

Mathematically, hyperbolic functions take a real argument without any inherent unit, similar to exponential functions. Many applied contexts still express x in radians or in the same units as an angle for comparison with trigonometric functions. This calculator lets you enter x as either a raw real number (interpreted as radians) or as degrees, which are internally converted to radians.

Why does the calculator show cosh²x − sinh²x?

The fundamental identity for hyperbolic functions is cosh²x − sinh²x = 1, analogous to cos²x + sin²x = 1 for circular trigonometric functions. Due to finite precision arithmetic, the computed value may differ slightly from 1, especially for large |x|, and the calculator shows this deviation to help diagnose numerical stability or extreme inputs.

Hyperbolic Functions Calculator – sinh, cosh, tanh (and reciprocals)

Compute sinh x, cosh x, tanh x and the reciprocal hyperbolic functions coth x, sech x, csch x for real arguments. Enter x in radians or in degrees and see a numerical check of the identity \(\cosh^2x - \sinh^2x = 1\).

Hyperbolic functions sinh · cosh · tanh coth · sech · csch

1. Hyperbolic functions calculator

Enter a real value for x, choose whether you are thinking in radians or degrees, and set your desired display precision. The tool computes \(\sinh x\), \(\cosh x\), \(\tanh x\) and the reciprocals \(\coth x\), \(\operatorname{sech} x\), \(\operatorname{csch} x\), plus a numerical check of the identity \(\cosh^2x - \sinh^2x = 1\).

Argument x

You can use either a dot or a comma as decimal separator. Very large |x| can lead to overflow in e^x.

Interpret x as

Radians (raw real x) Degrees (converted to radians)

Number format

Internally, all computations use full double-precision floating point. Only the displayed values are rounded for readability.

x (radians)

–

Internal argument used for e^x definitions

x (degrees)

–

For comparison with trigonometric angles

cosh²x − sinh²x

–

Should be ≈ 1; deviations are due to finite precision

Function	Value	Definition / note
sinh x		\(\sinh x = \dfrac{e^x - e^{-x}}{2}\)
cosh x		\(\cosh x = \dfrac{e^x + e^{-x}}{2}\)
tanh x		\(\tanh x = \dfrac{\sinh x}{\cosh x}\) (for \(\cosh x \neq 0\))
coth x		\(\coth x = \dfrac{\cosh x}{\sinh x}\) (for \(\sinh x \neq 0\))
sech x		\(\operatorname{sech} x = \dfrac{1}{\cosh x}\) (for \(\cosh x \neq 0\))
csch x		\(\operatorname{csch} x = \dfrac{1}{\sinh x}\) (for \(\sinh x \neq 0\))

Definitions of the hyperbolic functions

The hyperbolic functions are defined using the exponential function. For a real argument \(x\), the three primary hyperbolic functions are:

\[ \sinh x = \frac{e^x - e^{-x}}{2}, \qquad \cosh x = \frac{e^x + e^{-x}}{2}, \qquad \tanh x = \frac{\sinh x}{\cosh x}. \]

Their reciprocals are defined analogously to trigonometry:

\[ \coth x = \frac{\cosh x}{\sinh x}, \qquad \operatorname{sech} x = \frac{1}{\cosh x}, \qquad \operatorname{csch} x = \frac{1}{\sinh x}. \]

Key identity: \(\cosh^2x - \sinh^2x = 1\)

Just as trigonometric functions satisfy \(\cos^2\theta + \sin^2\theta = 1\), hyperbolic functions satisfy the identity

\[ \cosh^2x - \sinh^2x = 1 \]

This identity can be verified directly from the exponential definitions. Start from \(\cosh x = \tfrac{e^x + e^{-x}}{2}\) and \(\sinh x = \tfrac{e^x - e^{-x}}{2}\), compute the squares, and subtract. All cross terms cancel and the result is identically 1 for all real \(x\).

Hyperbolic functions vs trigonometric functions

Although the notations are similar, hyperbolic and circular trigonometric functions behave differently:

\(\sin x\) and \(\cos x\) are bounded between -1 and 1, whereas \(\sinh x\) and \(\cosh x\) grow roughly like \(\tfrac{1}{2}e^{|x|}\) for large \(|x|\).
Trigonometric functions are periodic; hyperbolic functions are not periodic.
Geometrically, \((\cos \theta, \sin \theta)\) parameterizes the unit circle \(x^2 + y^2 = 1\), while \((\cosh t, \sinh t)\) parameterizes the right branch of the unit hyperbola \(x^2 - y^2 = 1\).

Where do hyperbolic functions appear?

Hyperbolic functions arise naturally in many areas of mathematics and physics:

The shape of a hanging cable or chain (a catenary) is described by \(y = a \cosh(x/a)\).
Solutions of certain linear differential equations (e.g. for beams, heat flow, and wave equations) involve combinations of \(\sinh\) and \(\cosh\).
In special relativity, Lorentz transformations can be expressed using hyperbolic functions of the “rapidity” parameter.
They appear in complex analysis when relating trigonometric and exponential functions via Euler’s formula, and in expressions of inverse trigonometric functions.

FAQ – Using the hyperbolic functions calculator

What happens for very large |x|?

Since \(\sinh x\) and \(\cosh x\) contain terms \(e^x\) and \(e^{-x}\), they grow very quickly in magnitude as |x| increases. In double-precision floating point, \(\cosh x\) overflows to infinity for |x| beyond roughly 710. The calculator will show Infinity (or very large scientific notation) in such cases and the identity check will no longer be meaningful.

Why do coth x and csch x sometimes show “undefined”?

The reciprocals \(\coth x = \cosh x / \sinh x\) and \(\operatorname{csch} x = 1 / \sinh x\) are undefined when \(\sinh x = 0\). Numerically, the calculator treats values with very small |sinh x| as effectively zero and prints a clear “undefined (division by zero)” message to avoid misleading gigantic outputs.

Is there a geometric meaning for the input x?

For trigonometric functions, an angle \(\theta\) is naturally associated with arc length on the unit circle. For hyperbolic functions, the parameter \(x\) can be interpreted as a hyperbolic angle, related to the area of a sector of the unit hyperbola. In many applied settings, however, x simply plays the role of a real parameter without a direct geometric unit.

Can I invert these functions with this calculator?

This page focuses on the direct hyperbolic functions. Inverse hyperbolic functions (arsinh, arcosh, artanh, etc.) have well-known logarithmic formulas and are often required in integration and equation solving. For production use you would typically implement them with dedicated inverse hyperbolic calculators, where domain restrictions and branch choices are handled carefully.