Hyperbolic Cosine (cosh) Calculator

This hyperbolic cosine calculator works out cosh(x) for any real number x you enter, and shows sinh(x) and tanh(x) alongside so you can compare all three hyperbolic functions from a single input. The hyperbolic cosine is defined as cosh(x) = (e to the x, plus e to the minus x), divided by two, built directly from the exponential function rather than from angles like ordinary cosine. It is an even function, so cosh(-x) always equals cosh(x), and its smallest possible value is one, occurring at x = 0, with the curve rising symmetrically on either side. This shape matters beyond pure mathematics: a flexible cable, chain or rope hanging freely under its own weight naturally forms a catenary, whose equation is a scaled cosh curve, which is why the function turns up regularly in structural engineering, bridge and arch design, and architecture. To use the calculator, type your value of x into the single input field and the results update immediately, giving you cosh(x) as the main figure with sinh(x) and tanh(x) displayed as supporting values, all rounded for easy reading. A worked example is included below, showing that cosh(1) comes to about 1.543081, so you can check your own entries against a known result. It suits students, engineers and architects working with hyperbolic functions, catenary curves or exponential growth problems.

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Value of x

1.543081

sinh(x)1.175201

tanh(x)0.761594

The formula

The hyperbolic cosine is cosh(x) = (e^x + e^-x) / 2. It is an even function, cosh(-x) = cosh(x), with a minimum value of 1 at x = 0.

Worked example

cosh(1) = (e + 1/e) / 2, which is about 1.543081. Enter 1 above to confirm.

Frequently asked questions

What is the formula for cosh?

cosh(x) = (e^x + e^-x) / 2, built from the exponential function.

Why is cosh never less than one?

Because (e^x + e^-x) / 2 is smallest when x = 0, where it equals 1. It rises on both sides.

What is a catenary?

The curve a hanging cable forms under gravity. Its equation is a scaled cosh, which is why cosh appears in engineering.

Who this calculator is for

This calculator is for students, engineers and architects working with hyperbolic functions and catenaries.

What this calculator assumes

You enter a real number x.
cosh is computed from the exponential definition.
Results are rounded for display.

Formula and sources

Related calculators

Hyperbolic Sine Calculator: sinh(x).
Hyperbolic Tangent Calculator: tanh(x).
Inverse Hyperbolic Sine Calculator: arcsinh(x).

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