Inverse Hyperbolic Functions Calculator

Inverse Hyperbolic Functions:

\[ \text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1}) \] \[ \text{arcosh}(x) = \ln(x + \sqrt{x^2 - 1}) \]

Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Are Inverse Hyperbolic Functions?

Inverse hyperbolic functions are the inverse functions of the hyperbolic functions. They are useful in many areas of mathematics, physics, and engineering, particularly in solving certain types of differential equations and in describing special relativity.

2. How Does the Calculator Work?

The calculator uses the following formulas:

\[ \text{arsinh}(x) = \ln(x + \sqrt{x^2 + 1}) \] \[ \text{arcosh}(x) = \ln(x + \sqrt{x^2 - 1}) \]

Where:

\( x \) — Input value (unitless)
\( \ln \) — Natural logarithm (base e)
\( \sqrt{} \) — Square root function

Explanation: These functions are defined as the inverses of the hyperbolic sine and cosine functions, respectively.

3. Applications of Inverse Hyperbolic Functions

Details: Inverse hyperbolic functions appear in solutions of certain differential equations, in the description of catenary curves, in special relativity, and in the integration of certain functions.

4. Using the Calculator

Tips: Enter a numeric value for x, select the function you want to calculate (arsinh or arcosh), and click "Calculate". Note that arcosh(x) is only defined for x ≥ 1.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between arsinh and arcsin?
A: arcsin is the inverse of the trigonometric sine function, while arsinh is the inverse of the hyperbolic sine function. They are different functions with different properties.

Q2: Why is arcosh(x) only defined for x ≥ 1?
A: The hyperbolic cosine function cosh(x) has a minimum value of 1, so its inverse function arcosh(x) can only accept values ≥ 1.

Q3: What are the ranges of these functions?
A: arsinh(x) can take any real number and returns any real number. arcosh(x) takes x ≥ 1 and returns values ≥ 0.

Q4: Are there other inverse hyperbolic functions?
A: Yes, there are also artanh(x), arcoth(x), arsech(x), and arcsch(x), which are the inverses of the other hyperbolic functions.

Q5: How are these functions related to complex numbers?
A: Inverse hyperbolic functions can be expressed using complex logarithms and have interesting relationships with inverse trigonometric functions in the complex plane.