Math
Hyperbolic Functions Calculator
Hyperbolic functions are built from e^x and e^(−x) the way trigonometric functions are built from the unit circle.
Enter x and the hyperbolic functions calculator evaluates all six functions with their exponential definitions written out, or enter a value and get the inverse (asinh, acosh, atanh, and the rest) from its exact logarithmic form, with the domain checked first.
Any real number (|x| ≤ 700).
Try an example
Result
Value
—
- Exponential / logarithmic form
- —
- sinh
- —
- cosh
- —
- tanh
- —
- coth
- —
- sech
- —
More details (3 more)
- csch
- —
- Identity check
- —
- e^x and e^(−x)
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the hyperbolic functions calculator solves
sinh and cosh are the odd and even halves of e^x. Their ratio is tanh, and the The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5. give coth, sech, and csch. They describe hanging chains (catenary), special relativity, and solutions of many differential equations. Unlike sin and cos they are not periodic: cosh grows like e^x/2 and tanh flattens toward ±1.
Inverse hyperbolic functions
| Function | Logarithmic form | Domain |
|---|---|---|
| asinh y | ln(y + √(y² + 1)) | all real y |
| acosh y | ln(y + √(y² − 1)) | y ≥ 1 |
| atanh y | ½ ln((1 + y)/(1 − y)) | −1 < y < 1 |
| acoth y | ½ ln((y + 1)/(y − 1)) | |y| > 1 |
| asech y | ln((1 + √(1 − y²))/y) | 0 < y ≤ 1 |
| acsch y | ln(1/y + √(1/y² + 1)) | y ≠ 0 |
How to use it
- Choose a hyperbolic function and enter x, or switch to inverse mode and enter y.
- Read the value and the exponential (or logarithmic) form that produced it.
- All six hyperbolic values at the resulting x are listed, plus the identity check.
How to read the answer
sinh is odd and unbounded, cosh is even with minimum 1 at x = 0, and tanh is odd and squeezed between −1 and 1. coth and csch are undefined at x = 0. Inverse functions return the x whose hyperbolic value is y; outside the listed domains they are undefined and the calculator says so.
Common mistakes and edge cases
- Using sin and cos identities unchanged: the hyperbolic identity is cosh² − sinh² = 1, with a minus sign.
- Asking for acosh of a number below 1 or atanh of a number outside (−1, 1); both are undefined for real numbers.
- Expecting periodic behaviour: hyperbolic functions never repeat.
- Evaluating coth or csch at 0, where sinh is 0.
Worked examples
sinh(1)
≈ 1.1752012
Value
1.17520119
cosh(0)
= 1, the minimum of cosh
Value
1
tanh of a negative number
tanh(−2) ≈ −0.9640
Value
-0.96402758
Inverse: asinh(2)
ln(2 + √5) ≈ 1.4436
Value
1.44363548
Inverse: atanh(0.5)
½ ln 3 ≈ 0.5493
Value
0.54930614
Large x
cosh(10) ≈ 11013.23
Value
11,013.2329201
Outside the domain
acosh(0.5) is undefined
Value
Error
Frequently asked questions
What are hyperbolic functions?+
Combinations of e^x and e^(−x): sinh x = (e^x − e^(−x))/2 and cosh x = (e^x + e^(−x))/2, with tanh, coth, sech, and csch defined from them like the trig ratios. They parametrise the hyperbola x² − y² = 1 the way sin and cos parametrise the circle.
How is cosh related to a hanging chain?+
A chain hanging under its own weight takes the shape y = a·cosh(x/a), the catenary. That is why cosh appears in cables and arches.
Why is atanh only defined between −1 and 1?+
tanh never leaves the interval (−1, 1), so no real x has tanh x = 2. The logarithmic formula would need ln of a negative number.
What is the hyperbolic identity?+
cosh² x − sinh² x = 1. Compare the circular identity cos² x + sin² x = 1; the sign flips because of the minus in sinh.
Are sinh and cosh periodic?+
No. Over the real numbers they grow without bound as |x| increases. Only over complex arguments do they show periodicity (with period 2πi).
How do I compute asinh by hand?+
Use asinh y = ln(y + √(y² + 1)). For y = 2: ln(2 + √5) ≈ ln(4.236) ≈ 1.4436.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
References
- OpenStax, Calculus Volume 1, Section 6.9 “Calculus of the Hyperbolic Functions” — Definitions, identities, and inverse hyperbolic functions in logarithmic form.
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Last updated: September 4, 2026