Inverse hyperbolic functionsEdexcel International A Level Further Maths: Mind map

Exam questions on Inverse hyperbolic functions

1. The inverse hyperbolic functions have the logarithmic forms $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$ for all real $x$, $\operatorname{arcosh}x=\ln\left(x+\sqrt{x^2-1}\right)$ for $x\ge1$, and $\operatorname{artanh}x=\frac12\ln\frac{1+x}{1-x}$ for $|x|<1$.
   Find the exact value of $\operatorname{arcosh}5$, giving your answer in the form $\ln\left(a+b\sqrt6\right)$.2 marks
2. The function $f$ is defined by $f(x)=\operatorname{arcosh}x$.
   Explain how the graph of $y=\operatorname{arcosh}x$ is related to the graph of $y=\cosh x$.2 marks
3. For real $x$, $y=\operatorname{arsinh}x$ means that $x=\sinh y$.
   Prove that $\operatorname{arsinh}x=\ln\left(x+\sqrt{x^2+1}\right)$.3 marks