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