Inverse Hyperbolic Functions Calculator
\[ \operatorname{arcsinh}{x},\operatorname{arccosh}{x},\operatorname{arctanh}{x} \]
- Integer, decimal, and fraction can be entered.
- Enter fraction as 1/2, 3/5, etc.
- Hyperbolic Functions Calculator is available at this link.
What are inverse hyperbolic functions?
Inverse hyperbolic functions are the inverse of hyperbolic functions.
Inverse hyperbolic functions are defined as follows
\begin{align} \operatorname{arcsinh}{x}&=\ln{\left(x+\sqrt{x^{2}+1}\right)}\notag\\[+10pt] \operatorname{arccosh}{x}&=\ln{\left(x+\sqrt{x^{2}-1}\right)}\notag\\[+10pt] \operatorname{arctanh}{x}&=\frac{1}{2}\ln{\frac{1+x}{1-x}}\notag\\ \end{align}
When dealing with inverse hyperbolic functions in the real number range, there is a range of definitions as follows.
This site performs calculations in the complex number range.
| Inverse Hyperbolic Functions | Domain |
|---|---|
| \( \operatorname{arcsinh}{x} \) | \( x\in\mathbb{R} \) |
| \( \operatorname{arccosh}{x} \) | \( x\geq1 \) |
| \( \operatorname{arctanh}{x} \) | \( -1\lt x\lt1 \) |
Example
Example 1
Calculate \( \operatorname{arcsinh}{0} \).
Solution
\begin{align} \operatorname{arcsinh}{0}&=\ln{\left(0+\sqrt{0^{2}+1}\right)}\notag\\ &=\ln{1}\notag\\ &=0\notag \end{align}
Example 2
Calculate \( \operatorname{arccosh}{1} \).
Solution
\begin{align} \operatorname{arccosh}{1}&=\ln{\left(1+\sqrt{1^{2}-1}\right)}\notag\\ &=\ln{1}\notag\\ &=0\notag \end{align}
例題３
Calculate \( \operatorname{arctanh}{0} \).
解答
\begin{align} \operatorname{arctanh}{0}&=\frac{1}{2}\ln{\frac{1+0}{1-0}}\notag\\[+10pt] &=\frac{1}{2}\ln{1}\notag\\[+10pt] &=0\notag \end{align}