Hyperbolic Functions Calculator
Calculate hyperbolic sine, cosine, tangent, and their inverses.
Result
—
Enter value and select function
sinh(x)
—
cosh(x)
—
tanh(x)
—
Domain
—
Hyperbolic Functions
Hyperbolic functions are analogs of trigonometric functions, defined using the hyperbola rather than the circle.
Definitions
sinh(x) = (e^x - e^(-x)) / 2
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
cosh(x) = (e^x + e^(-x)) / 2
tanh(x) = sinh(x) / cosh(x)
Key Properties
| Function | Domain | Range |
|---|---|---|
| sinh(x) | All real | All real |
| cosh(x) | All real | [1, ∞) |
| tanh(x) | All real | (-1, 1) |
| asinh(x) | All real | All real |
| acosh(x) | [1, ∞) | [0, ∞) |
| atanh(x) | (-1, 1) | All real |
Identities
cosh²(x) - sinh²(x) = 1
This is the hyperbolic analog of the Pythagorean identity.
Applications
- Catenary curve (sagging cables)
- Special relativity
- Electrical engineering
- Complex number trigonometry
Frequently Asked Questions
Why are they called hyperbolic?
The point (cosh(t), sinh(t)) lies on the hyperbola x² - y² = 1, just as (cos(t), sin(t)) lies on the unit circle.
What is tanh(x) used for?
Tanh is used in machine learning as an activation function, mapping any input to (-1, 1). It's also used in special relativity for rapidity.