Answer
$cos\theta= -\frac{\sqrt 5}{3}$,
$tan\theta= -\frac{2\sqrt 5}{5}$,
$cot\theta= -\frac{\sqrt 5}{2}$,
$sec\theta= -\frac{3\sqrt 5}{5}$,
$csc\theta= \frac{3}{2}$.
Work Step by Step
Given $sin\theta=\frac{2}{3}$, let $y=2, r=3$, we can get $x=\sqrt {r^2-y^2}=\sqrt 5$. Use the fact that $tan\theta\lt0,sin\theta\gt0$, thus $\theta$ is in quadrant II to determine the sign of each function:
$cos\theta=-\frac{x}{r}=-\frac{\sqrt 5}{3}$,
$tan\theta=-\frac{y}{x}=-\frac{2\sqrt 5}{5}$,
$cot\theta=-\frac{x}{y}=-\frac{\sqrt 5}{2}$,
$sec\theta=-\frac{r}{x}=-\frac{3\sqrt 5}{5}$,
$csc\theta=\frac{r}{y}=\frac{3}{2}$.
