Answer
$sin\theta= -\frac{2\sqrt {2}}{3}$,
$cos\theta= \frac{1}{3}$,
$tan\theta= -2\sqrt 2$,
$cot\theta= -\frac{\sqrt {2}}{4}$,
$csc\theta= -\frac{3\sqrt {2}}{4}$.
Work Step by Step
Given $sec\theta=3$, let $x=1, r=3$. We have $y=\sqrt {r^2-x^2}=2\sqrt {2}$. As $ \theta $ is in quadrant IV, the signs of the other functions are also known. We have:
$sin\theta=-\frac{y}{r}=-\frac{2\sqrt {2}}{3}$,
$cos\theta=\frac{x}{r}=\frac{1}{3}$,
$tan\theta=-\frac{y}{x}=-2\sqrt 2$,
$cot\theta=-\frac{x}{y}=-\frac{\sqrt {2}}{4}$,
$sec\theta=\frac{r}{x}=3$,
$csc\theta=-\frac{r}{y}=-\frac{3\sqrt {2}}{4}$.
