\[\begin{aligned}
\sinh^{-1}(x) &= \ln(x+\sqrt{x^2+1}) \\
\cosh^{-1}(x) &= \ln(x+\sqrt{x^2-1}) \\
\tanh^{-1}(x) &= \frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)
\end{aligned}\]
Variables
asinhinverse hyperbolic sine value
acoshinverse hyperbolic cosine value
atanhinverse hyperbolic tangent value
xindependent variable
Description
What is this formula?
These formulas calculate inverse hyperbolic functions using logarithmic expressions. They determine the hyperbolic angle corresponding to a given value.
When to use it
Use these formulas when solving equations involving hyperbolic functions or modeling logarithmic-hyperbolic relationships.
Example
For:
x=2
asinh=ln(2+sqrt(5))
asinh≈1.444
Applications
Advanced calculus
Differential equations
Electrical engineering
Relativity physics
Signal analysis
Mathematical modeling