Answer
$\ln(2-\sqrt{3})$
Work Step by Step
Substitute into the relevant formula:
$\displaystyle \mathrm{c}\mathrm{s}\mathrm{c}\mathrm{h}^{-1}(-\frac{1}{\sqrt{3}})=\ln[\frac{1}{-\frac{1}{\sqrt{3}}}+\frac{\sqrt{1+(-\frac{1}{\sqrt{3}}})^{2}}{|-\frac{1}{\sqrt{3}}|}]$
$=\displaystyle \ln[-\sqrt{3}+\frac{\sqrt{4/3}}{\frac{1}{\sqrt{3}}}]$
$=\ln[-\sqrt{3}+\sqrt{3}\sqrt{4/3}]$
$=\ln[-\sqrt{3}+\sqrt{4}]$
$=\ln(2-\sqrt{3})$
