Math Doubts

Inverse Hyperbolic functions

The inverse form of a hyperbolic function is called the inverse hyperbolic function.

Hyperbolic functions are six types. So, the inverse hyperbolic functions are also six types. Each hyperbolic function is defined in exponential functions form. So, each inverse hyperbolic function is defined in logarithmic function form.

List of functions

Here is the list of six inverse hyperbolic functions in logarithmic functions form with proofs for beginners.


Inverse Hyperbolic Sine Function

$\large \sinh^{-1}{x} \,=\, \log_{e}{(x+\sqrt{x^2+1})}$


Inverse Hyperbolic Cosine Function

$\large \cosh^{-1}{x} \,=\, \log_{e}{(x+\sqrt{x^2-1})}$


Inverse Hyperbolic Tangent Function

$\large \tanh^{-1}{x} \,=\, \dfrac{1}{2}\log_{e}{\Bigg(\dfrac{1+x}{1-x}\Bigg)}$


Inverse Hyperbolic Cotangent Function

$\large \coth^{-1}{x} \,=\, \dfrac{1}{2}\log_{e}{\Bigg(\dfrac{x+1}{x-1}\Bigg)}$


Inverse Hyperbolic Secant Function

$\large {\mathop{\rm sech}\nolimits}^{-1}{x} = \log{\Bigg(\dfrac{1+\sqrt{1-x^2}}{x}\Bigg)}$


Inverse Hyperbolic Cosecant Function

${\mathop{\rm csch}\nolimits}^{-1}{x} = \log{\Bigg(\dfrac{1-\sqrt{1+x^2}}{x}\Bigg)}$

Math Doubts
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more
Follow us on Social Media
Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.

Learn more