Answer
$sin\theta=-\frac{\sqrt 5}{3}$,
$tan\theta=-\frac{\sqrt 5}{2}$,
$cot\theta=-\frac{2\sqrt 5}{5}$,
$sec\theta=\frac{3}{2}$,
$csc\theta=-\frac{3\sqrt 5}{5}$.
Work Step by Step
Given $cos\theta=\frac{2}{3}$ and $\theta$ in quadrant IV, let $x=2, r=3$, we have $y=-\sqrt {3^2-2^2}=-\sqrt 5$ and
$sin\theta=\frac{y}{r}=-\frac{\sqrt 5}{3}$,
$tan\theta=\frac{y}{x}=-\frac{\sqrt 5}{2}$,
$cot\theta=\frac{1}{tan\theta}=-\frac{2\sqrt 5}{5}$,
$sec\theta=\frac{1}{cos\theta}=\frac{3}{2}$,
$csc\theta=\frac{1}{sin\theta}=-\frac{3\sqrt 5}{5}$.
