Cosh Calculator - Hyperbolic Cosine
Enter any real number x to compute cosh(x) and all six hyperbolic functions at once. Switch to inverse mode to go backwards: enter a known cosh value (1 or above) and recover x via arcosh. Every result comes with a step-by-step derivation showing the exponential definition.
Formula
Worked example
For x = 1: e^1 = 2.71828, e^-1 = 0.36788. cosh(1) = (2.71828 + 0.36788) / 2 = 1.54308. sinh(1) = (2.71828 - 0.36788) / 2 = 1.17520. Identity check: 1.54308^2 - 1.17520^2 = 2.38109 - 1.38109 = 1.00000.
What is the hyperbolic cosine function?
The hyperbolic cosine, written cosh(x), is defined as (e^x + e^(-x)) / 2. It is one of six hyperbolic functions that are the hyperbolic counterparts of the trigonometric functions. While the ordinary cosine traces a circle, cosh traces one side of a hyperbola: the point (cosh t, sinh t) always lies on the unit hyperbola x^2 - y^2 = 1. This Pythagorean-style identity, cosh^2(x) - sinh^2(x) = 1, is the hyperbolic analog of the familiar sin^2 + cos^2 = 1 identity from trigonometry.
Key properties of cosh
The cosh function has several distinctive properties. Its domain is all real numbers and its range is [1, infinity): it never drops below 1. cosh(0) = 1 is the global minimum. The function is even: cosh(-x) = cosh(x), so its graph is symmetric about the y-axis. It is not periodic (unlike the circular cosine), and it grows without bound as |x| increases, approaching e^|x| / 2 for large values. The derivative of cosh(x) is sinh(x), and the derivative of sinh(x) is cosh(x) -- these two functions differentiate into each other, unlike the circular trig pair which introduces a sign flip.
The catenary: cosh in the real world
The most famous physical application of cosh is the catenary curve, the shape a uniform cable takes under its own weight when suspended from two points. The equation is y = a * cosh(x / a), where a is the ratio of tension to linear mass density. This is why power lines, suspension bridge cables, and hanging chains follow the cosh curve. The St. Louis Gateway Arch is also designed as an inverted catenary, for maximum structural efficiency. The word catenary comes from the Latin catena, meaning chain.
The six hyperbolic functions and their inverses
Alongside cosh(x) and sinh(x), the full set of six hyperbolic functions is: tanh(x) = sinh / cosh (hyperbolic tangent, always between -1 and 1), coth(x) = cosh / sinh (hyperbolic cotangent, defined for x not 0), sech(x) = 1 / cosh (hyperbolic secant), and csch(x) = 1 / sinh (hyperbolic cosecant, defined for x not 0). Each has an inverse: arcosh(y) = ln(y + sqrt(y^2 - 1)) for y >= 1; arsinh(y) = ln(y + sqrt(y^2 + 1)) for all y; artanh(y) = (1/2) * ln((1 + y) / (1 - y)) for |y| < 1. This calculator computes all six at once for any x, and can also run the inverse to recover x from a known cosh value.
Notable cosh values
| x | cosh(x) | sinh(x) | tanh(x) | Notes |
|---|---|---|---|---|
| -2 | 3.762196 | -3.626860 | -0.964028 | Symmetric with x = 2 |
| -1 | 1.543081 | -1.175201 | -0.761594 | Symmetric with x = 1 |
| 0 | 1.000000 | 0.000000 | 0.000000 | Global minimum of cosh |
| 0.5 | 1.127626 | 0.521095 | 0.462117 | |
| 1 | 1.543081 | 1.175201 | 0.761594 | tanh(1) approx 0.762 |
| ln(2) approx 0.693 | 1.250000 | 0.750000 | 0.600000 | cosh(ln 2) = 5/4 exactly |
| 1.317 | 2.000000 | 1.732051 | 0.866025 | cosh = 2, sinh = sqrt(3) |
| 2 | 3.762196 | 3.626860 | 0.964028 | tanh approaching 1 |
| 3 | 10.067662 | 10.017875 | 0.995055 | Growth becomes near-exponential |
| 5 | 74.209930 | 74.203265 | 0.999909 | tanh very close to 1 |
Exact or rounded values of cosh at commonly referenced points. The minimum value of cosh is 1, at x = 0.
Frequently asked questions
What does cosh(x) mean?
cosh(x) is the hyperbolic cosine of x, defined as (e^x + e^(-x)) / 2. It is named by analogy with the circular cosine but uses exponentials rather than angles. Its minimum value is 1 (at x = 0) and it grows symmetrically in both directions.
What is the difference between cos and cosh?
cos(x) is the circular (trigonometric) cosine, which oscillates between -1 and 1 with period 2*pi. cosh(x) is the hyperbolic cosine, which starts at 1 when x = 0 and grows unboundedly in both directions -- it never oscillates. Both satisfy Pythagorean identities: sin^2 + cos^2 = 1 for circular, cosh^2 - sinh^2 = 1 for hyperbolic (note the minus sign).
What is the inverse of cosh?
The inverse is arcosh (or cosh^-1), defined as arcosh(y) = ln(y + sqrt(y^2 - 1)). It is only valid for y >= 1, because cosh always produces values >= 1. The output of arcosh is non-negative by convention (since cosh is even, there are always two x values with the same cosh -- the positive one is returned).
Why does cosh appear in engineering and physics?
Cosh appears naturally in problems involving exponential growth in both directions: catenary cables, electric field distributions, heat flow, and solutions to certain differential equations. Special relativity uses hyperbolic functions to describe Lorentz boosts, and cosh appears in waveguide equations, beam deflection problems, and probability distributions such as the hyperbolic secant distribution.
Is cosh an even or odd function?
Cosh is an even function: cosh(-x) = cosh(x) for all x. This means the graph is symmetric about the y-axis. By contrast, sinh is an odd function: sinh(-x) = -sinh(x), giving a rotationally symmetric graph about the origin.
What is the Pythagorean identity for hyperbolic functions?
The fundamental identity is cosh^2(x) - sinh^2(x) = 1 for all real x. This is analogous to sin^2(x) + cos^2(x) = 1, with the key difference being the minus sign rather than a plus sign. Dividing through by cosh^2 gives 1 - tanh^2(x) = sech^2(x), and dividing by sinh^2 gives coth^2(x) - 1 = csch^2(x).