Answer
$sin\theta= \frac{1}{3}$,
$cos\theta= -\frac{2\sqrt {2}}{3}$,
$tan\theta= -\frac{\sqrt {2}}{4}$,
$cot\theta= -2\sqrt 2$,
$sec\theta= -\frac{3\sqrt {2}}{4}$.
Work Step by Step
Given $csc\theta=3$, let $r=3, y=1$, we can get $x=\sqrt {r^2-y^2}=2\sqrt {2}$. Use the fact that $cot\theta\lt0,sin\theta=\frac{1}{csc\theta}\gt0\longrightarrow \theta$ is in quadrant II to determine the sign of each function:
$sin\theta=\frac{y}{r}=\frac{1}{3}$,
$cos\theta=-\frac{x}{r}=-\frac{2\sqrt {2}}{3}$,
$tan\theta=-\frac{y}{x}=-\frac{\sqrt {2}}{4}$,
$cot\theta=-\frac{x}{y}=-2\sqrt 2$,
$sec\theta=-\frac{r}{x}=-\frac{3\sqrt {2}}{4}$.
