Answer
$\displaystyle \frac{2\sqrt{5}}{5}$
Work Step by Step
We don't have a tabular angle for which $\displaystyle \sin t=\frac{2}{3}$, so we construct a right triangle in which this trigonometric ratio (opposite/hypotenuse) equals $\displaystyle \frac{2}{3}$. See image below.
$t=\displaystyle \sin^{-1}\frac{2}{3}.$
$\tan t$ = (opposite leg)/(adjacent leg)
we find the adjacent leg using the Pythagorean th.: $\left[\begin{array}{l}
a^{2}+2^{2}=3^{2}\\
a=\sqrt{9-4}\\
a=\sqrt{5}
\end{array}\right]$
$\displaystyle \tan t=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$
