What is the value of $\tan({\frac{1}{2}}\sin^{-1}{\frac{3}{4}})$?
$\begin{array}{1 1} \frac{-4+\sqrt 7}{3} \\ \frac{-4 - \pi}{3} \\ \frac{4+\sqrt 7}{3} \\ \frac{4- \sqrt 7}{3} \end{array}$