Basics of Inverse Hyperbolic Functions

Prove the identity ${\sinh }^{-1}\left(x\right)=\log \left(x+\sqrt{x^2+1}\right), x\ge 1$.

If ${\coth }^{-1}\left(x\right)=\frac{1}{2}\ln \left(\frac{x+k}{x-k}\right), \left|x\right|>1$, then find the value of $k$.

Find the value of ${\mathrm{sech}}^{-1}\left(\frac{1}{3}\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\mathrm{sech}}^{-1}\left(1\right)$.

Find the value of ${\mathrm{sech}}^{-1}\left(\frac{1}{2}\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Prove the identity ${\mathrm{sech}}^{-1}\left(x\right)=\ln \left(\frac{1+\sqrt{1-x^2}}{x}\right), \left(0<x\le 1\right)$.

Find the value of ${\mathrm{cosech}}^{-1}\left(4\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\mathrm{cosech}}^{-1}\left(3\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\mathrm{cosech}}^{-1}\left(2\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\mathrm{cosech}}^{-1}\left(1\right)$.

[Use calculator to find the approximate value of the logarithmic value].

Prove the identity ${\mathrm{cosech}}^{-1}\left(x\right)=\ln \left(\frac{1}{x}+\frac{\sqrt{1+x^{\mathit{2}}}}{x}\right), x\ne 0$

Find the value of ${\coth }^{-1}\left(4\right)$. [Write the answer up to two decimal place]

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\coth }^{-1}\left(3\right)$. [Write the answer up to two decimal place]

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\coth }^{-1}\left(-2\right)$. [Write the answer up to two decimal place]

[Use calculator to find the approximate value of the logarithmic value].

Prove the identity ${\coth }^{-1}\left(x\right)=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right), \left|x\right|>1$.

Find the value of ${\tanh }^{-1}\left(0\right)$.  

Find the value of ${\tanh }^{-1}\left(\frac{-1}{2}\right)$.  [Write the answer up to two decimal place]

[Use calculator to find the approximate value of the logarithmic value].

Prove the identity ${\tanh }^{-1}\left(x\right)=\frac{1}{2}\ln \left(\frac{1+x}{1-x}\right), \left|x\right|<1$.

Find the value of ${\cosh }^{-1}\left(9\right)$. [Write the answer up to two decimal place.]

[Use calculator to find the approximate value of the logarithmic value].

Find the value of ${\cosh }^{-1}\left(3\right)$. [Write the answer up to two decimal place.]

[Use calculator to find the approximate value of the logarithmic value].