About this calculator
Inverse hyperbolic function calculator is used to calculate inverse hyperbolic function values such as asinh, acosh, atanh, etc. The inverse hyperbolic function is the inverse function of the hyperbolic function and is commonly used in advanced mathematics, differential equations, integral transformations, relativistic models and engineering curve analysis.
Common formulas include asinh(x)=ln(x+√(x²+1)), acosh(x)=ln(x+√(x²-1)), atanh(x)=1/2·ln((1+x)/(1-x)). These formulas relate inverse hyperbolic functions to natural logarithms and are therefore very useful in integral and analytical calculations.
Different inverse hyperbolic functions have different domains: asinh is defined for all real numbers, acosh requires x ≥ 1, and atanh requires -1 < x < 1. Use this tool to quickly check whether the input is within the valid range and obtain the function value.
What it calculates
The inverse hyperbolic functions calculator evaluates asinh, acosh, atanh, acoth, asech, and acsch, helping recover the original input from a hyperbolic function value.
Formula
- asinh(x) = ln(x + sqrt(x^2 + 1)).
- acosh(x) = ln(x + sqrt(x^2 - 1)), with domain x >= 1.
- atanh(x) = 1/2 ln((1 + x) / (1 - x)), with domain -1 < x < 1.
Inputs
- Input value x.
- The inverse hyperbolic function to evaluate.
- Check whether the input lies in the real domain of that function.
Example
| Input | Function | Note |
|---|---|---|
| x = 0 | asinh(x) | Result is 0 |
| x = 1 | acosh(x) | Result is 0 |
| x = 0 | atanh(x) | Result is 0 |
| x = 2 | acosh(x) | Valid real input |
How to interpret the result
An inverse hyperbolic result is the value that produces the input through the corresponding hyperbolic function. For example, y = asinh(x) means sinh(y) = x.
Common mistakes
- Real acosh(x) requires x >= 1.
- Real atanh(x) requires -1 < x < 1.
- Inverse hyperbolic functions are not reciprocal functions; asinh(x) is not 1/sinh(x).
How to use
Start by selecting the inverse hyperbolic function to evaluate, such as asinh, acosh, or atanh. Then enter the value of variable x and click "Calculate" to get the result.
When calculating asinh(2), you can directly enter 2, and the result is equivalent to ln(2+√5). When evaluating acosh(3), the input must be greater than or equal to 1. When calculating atanh(0.5), the input must be between -1 and 1.
If the result looks large or the prompt is invalid, check the function domain first. Although inverse hyperbolic functions are similar in form to inverse trigonometric functions, their images, definition domains, and value ranges are different.
Main features
Supports common functions such as inverse hyperbolic sine, inverse hyperbolic cosine, and inverse hyperbolic tangent.
Determine whether the input is valid based on the function domain, suitable for advanced mathematics, calculus, integral simplification and engineering model calculations.
Shows the relationship between the inverse hyperbolic function and the natural logarithm formula, which can be used for quick value checking and learning verification.
Use cases
Inverse hyperbolic functions often appear in integral tables, for example ∫dx/√(x²+a²) is related to asinh and ∫dx/(1-x²) is related to atanh. When learning calculus, they can help identify standard integral forms.
In engineering and physics, hyperbolic functions and their inverse functions are used in catenaries, relativistic velocity transformations, some diffusion models, and nonlinear system analysis.
In data modeling, atanh is also commonly used in Fisher z transformation to handle statistical inference of correlation coefficients.