Hyperbolic cosine calculator
Enter a number x and get cosh(x), with the formula it comes from.
Decimals may use a point or a comma. A sum works too, such as 2^10 or sqrt(2).
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
Definition
cosh(x) = (eˣ + e⁻ˣ) / 2. It is always at least 1, with cosh(0) = 1, and it is symmetric: cosh(−x) = cosh(x).
The catenary
A cable or chain hanging freely between two points takes the shape y = a · cosh(x / a), called a catenary. That is why cosh appears in the design of bridges, power lines and arches. See also sinh and tanh.
Values to compare with
| x | cosh(x) |
|---|---|
| 0 | 1 |
| 0.5 | 1.1276 |
| 1 | 1.5431 |
| 2 | 3.7622 |
| 5 | 74.2099 |
A worked catenary
A chain hangs between two poles in the shape y = 2 · cosh(x / 2) with x in metres, measured from the lowest point. At the lowest point (x = 0) the height is 2. At 2 m to the side, it has risen to 2 × cosh(1) ≈ 3.09 m. Doubling the distance to 4 m gives 2 × cosh(2) ≈ 7.52 m.
cosh and sinh together
For large x, cosh(x) and sinh(x) are almost the same, since e⁻ˣ vanishes. The difference is clearest near zero, where cosh is 1 and sinh is 0. The gap is exactly e⁻ˣ, which you can check with sinh and the exponent calculator.
Frequently asked questions
What is the smallest value of cosh?
1, at x = 0. Cosh is symmetric and never drops below 1, in contrast to cos, which swings between −1 and 1.
What is the difference between cosh and cos?
cos(x) is the circular function that repeats every 360°. cosh(x) is built from exponentials, is never below 1, and grows without limit.
How does cosh relate to e^x?
cosh(x) is the average of eˣ and e⁻ˣ. Together with sinh it gives eˣ = cosh(x) + sinh(x).
How do I get the height of a hanging cable?
For a cable shaped like y = a · cosh(x / a), enter x / a into the calculator and multiply the result by a. With a = 2 and x = 2, 2 × cosh(1) ≈ 3.086.