In mathematics, the inverse hyperbolic functions are inverse functions of the hyperbolic function. For a given hyperbolic function, the size of hyperbolic angle is always equal to the area of some hyperbolic sector where x*y = 1 or it could be twice the area of corresponding sector for the hyperbola unit – x2 − y2 = 1, in the same way like the circular angle is twice the area of circular sector of the unit circle. In some case, the inverse hyperbolic functions are also named as area functions to realize the values of hyperbolic angles.
\[\large arsinh\;x=ln(x+\sqrt {x^{2}+1}\]
Inverse hyperbolic cosine (if the domain is the closed interval $(1, +\infty )$.
\[\large arcosh\;x=ln(x+\sqrt{x^{2}-1})\]
Inverse hyperbolic tangent [if the domain is the open interval (−1, 1)]
\[\large arcosh\;x=\frac{1}{2}\;ln\left(\frac{1+x}{1-x} \right )\]
Inverse hyperbolic cotangent [if the domain is the union of the open intervals (−∞, −1) and (1, +∞)]
\[\large arcosh\;x=\frac{1}{2}\;ln\left(\frac{x+1}{x-1} \right )\]
Inverse hyperbolic cosecant (if the domain is the real line with 0 removed)
\[\large arcosh\;x=ln\left(\frac{1}{x}+\sqrt{\frac{1}{x^{2}+1}}\right)=ln\left(\frac{1+\sqrt{x^{2}+1}}{x}\right)\]
Inverse hyperbolic secant (if the domain is the semi-open interval 0, 1)
\[\large arcosh\;x=ln\left(\frac{1}{x}+\sqrt{\frac{1}{x^{2}+1}}\right)=ln\left(\frac{1+\sqrt{1-x^{2}}}{x}\right)\]
Derivatives formula of Inverse Hyperbolic Functions
\[\large \frac{d}{dx}sinh^{-1}x=\frac{1}{\sqrt{x^{2}+1}}\]
\[\large \frac{d}{dx}cosh^{-1}x=\frac{1}{\sqrt{x^{2}-1}}\]
\[\large \frac{d}{dx}tanh^{-1}x=\frac{1}{1-x^{2}}\]
\[\large \frac{d}{dx}coth^{-1}x=\frac{1}{1-x^{2}}\]
\[\large \frac{d}{dx}sech^{-1}x=\frac{-1}{x\sqrt{1-x^{2}}}\]
\[\large \frac{d}{dx}csch^{-1}x=\frac{-1}{|x|\sqrt{1-x^{2}}}\]
The concept is not new but inverse hyperbolic functions exist in various differential equations in hyperbolic geometry or Laplace equations. These equations are important for the calculation variables in different sectors like physics, chemistry, heat transfer, electromagnetic theory, relativity theory, or fluid dynamics etc.
However, there are standard abbreviations used in mathematics for inverse hyperbolic functions and when they are combined together in logical form, it will make the formula. This would be easy understanding formula if you know its nomenclature and the basic knowledge of trigonometric functions too.
There are particular notations for the inverse function but it should be misunderstood by -1 in mathematics. It is just a shorthand practice for writing the inverse functions. The next term is hyperbolic functions whose numerator and denominator are arranged at the degree of two solve in terms of ex with the help of quadratic formula. Now take the natural logarithms to derive the natural expressions for inverse hyperbolic functions.