Answer
$\csc\theta=-3$
$\sin\theta=-\frac{1}{3}$
$ \cos\theta=\frac{2\sqrt 2}{3}$
$\sec\theta=\frac{3\sqrt 2}{4}$
$\tan\theta=-\frac{\sqrt 2}{4}$
$\cot\theta=-2\sqrt 2$
Work Step by Step
Given $\csc\theta=-3$
$\sin\theta=\frac{1}{\csc\theta}=\frac{1}{-3}=-\frac{1}{3}$
Recall: $\sin^{2}\theta+\cos^{2}\theta=1$
$\implies (-\frac{1}{3})^{2}+\cos^{2}\theta=1$
Or $\cos^{2}\theta=1-\frac{1}{9}=\frac{8}{9}$
$\implies \cos\theta=\pm\frac{\sqrt 8}{\sqrt 9}=\frac{2\sqrt 2}{3}$ (given $\cos\theta\gt0$)
$\sec\theta=\frac{1}{\cos\theta}=\frac{3}{2\sqrt 2}=\frac{3\sqrt 2}{4}$
$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{1}{3}}{\frac{2\sqrt 2}{3}}=-\frac{1}{2\sqrt 2}=-\frac{\sqrt 2}{4}$
$\cot\theta=\frac{1}{\tan\theta}=-2\sqrt 2$
