Answer
$\sin\theta=-\frac{1}{2}$
$\cos\theta=-\frac{\sqrt 3}{2}$
$\sec\theta=-\frac{2\sqrt 3}{3}$
$\tan\theta=\frac{\sqrt 3}{3}$
$\cot\theta=\sqrt 3$
Work Step by Step
$\csc\theta=-2$
$\sin\theta=\frac{1}{\csc\theta}=\frac{1}{-2}=-\frac{1}{2}$
$\sin^{2}\theta+\cos^{2}\theta=1\implies\cos^{2}\theta=1-\sin^{2}\theta$
$=1-(-\frac{1}{2})^{2}=\frac{3}{4}$
$\cos\theta$ is negative in the third quadrant. Therefore,
$\cos\theta=-\sqrt {\frac{3}{4}}=-\frac{\sqrt 3}{2}$
$\sec\theta=\frac{1}{\cos\theta}=-\frac{2}{\sqrt 3}=-\frac{2\sqrt 3}{3}$
$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{1}{2}}{-\frac{\sqrt 3}{2}}=\frac{1}{\sqrt 3}=\frac{\sqrt 3}{3}$
$\cot\theta=\frac{1}{\tan\theta}=\frac{1}{\frac{1}{\sqrt 3}}=\sqrt 3$
