Cosh Calculator

Compute the hyperbolic cosine cosh(x) = (e^x + e^-x) / 2 for any real number x, along with the related sinh(x), tanh(x), and e^x values.

Quick Facts

Definition
cosh(x) = (e^x + e^-x) / 2
An even function: cosh(-x) = cosh(x), with minimum value cosh(0) = 1.
Core identity
cosh²(x) - sinh²(x) = 1
The hyperbolic counterpart of cos²θ + sin²θ = 1 for circular trig.

Your Results

Calculated
cosh(x)
-
Hyperbolic cosine
sinh(x)
-
(e^x - e^-x) / 2
tanh(x)
-
sinh(x) / cosh(x)
e^x
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Exponential term used above

Ready

Enter x and press Calculate.

How the Cosh Calculator works

This tool evaluates the hyperbolic cosine function, written cosh(x), for any real number x you enter. Alongside cosh(x) it also reports the two other core hyperbolic functions - sinh(x) and tanh(x) - plus the exponential term e^x that all three are built from.

Formula and method

The hyperbolic cosine is defined directly from the exponential function:

cosh(x) = (e^x + e^-x) / 2

where e is approximately 2.718281828, Euler's number. The two related functions shown alongside it are:

  • sinh(x) = (e^x - e^-x) / 2 - the hyperbolic sine
  • tanh(x) = sinh(x) / cosh(x) - the hyperbolic tangent, always between -1 and 1

These functions satisfy the identity cosh^2(x) - sinh^2(x) = 1, which is why the point (cosh(t), sinh(t)) traces out the hyperbola x^2 - y^2 = 1 as t varies - the same way (cos t, sin t) traces a circle. That parallel with circular trigonometry is where the name "hyperbolic" comes from.

Common sources of error

  • Confusing cosh with cos: the circular cosine cos(x) oscillates between -1 and 1; the hyperbolic cosine cosh(x) never oscillates and is always at least 1.
  • Overflow at large |x|: e^x grows so fast that cosh(x) exceeds the range double-precision numbers can represent once |x| is roughly 710 or more, so this calculator limits x to between -700 and 700.
  • Sign errors in sinh: sinh is an odd function (sinh(-x) = -sinh(x)), so flipping the sign of x flips the sign of sinh(x) but leaves cosh(x) unchanged.

Checking your result

A quick sanity check: cosh(x) should always be at least 1, and always at least as large as the absolute value of sinh(x). At x = 0, cosh should read exactly 1 and sinh exactly 0. For any x, squaring the reported cosh and sinh values and subtracting should land on 1 (cosh^2(x) - sinh^2(x) = 1), which is a reliable way to confirm the numbers are internally consistent.

Applications

The hyperbolic cosine describes the catenary curve of a hanging cable or chain under its own weight, appears in the equations of special relativity (rapidity) and electrical transmission lines, and is the real-axis counterpart of the circular cosine, since cos(ix) = cosh(x) for the imaginary unit i.

Frequently Asked Questions

What is the formula for cosh(x)?
The hyperbolic cosine is defined as cosh(x) = (e^x + e^-x) / 2, where e is Euler's number, approximately 2.718281828. It is the average of e^x and e^-x. Unlike the circular cosine, it is not periodic - it grows without bound as |x| increases in either direction.
How is cosh related to sinh and tanh?
sinh(x) = (e^x - e^-x) / 2 is the hyperbolic sine, and tanh(x) = sinh(x) / cosh(x) is the hyperbolic tangent. Together they satisfy the identity cosh^2(x) - sinh^2(x) = 1, the hyperbolic analogue of cos^2(theta) + sin^2(theta) = 1 for circular trigonometry.
Why is cosh(x) always 1 or greater?
cosh(0) = (e^0 + e^-0) / 2 = (1 + 1) / 2 = 1, which is the function's minimum. Since e^x and e^-x are always positive and their product is always 1, their average is smallest when the two terms are equal, at x = 0 - so cosh(x) never drops below 1 for any real x.
Where does cosh(x) show up in practice?
The hyperbolic cosine describes the shape of a hanging chain or cable under gravity, known as a catenary curve. It also appears in special relativity, in the equations for electrical transmission lines, and as the real part of cos(ix) in complex analysis, since cosh(x) = cos(ix).

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