Answer
$\tan{\theta}=-2 \sqrt{2}$
$\csc{\theta} =\dfrac{3\sqrt{2}}{4}$
$\sec{\theta} =-3$
$\cot{\theta} =-\dfrac{\sqrt{2}}{4}$
Work Step by Step
$\tan{\theta}= \dfrac{\sin{\theta}}{\cos{\theta}}$
$\therefore \tan{\theta}= \dfrac{\dfrac{2\sqrt{2}}{3}}{-\dfrac{1}{3}}=-2 \sqrt{2}$
$\csc{\theta} = \dfrac{1}{\sin{\theta}}$
$\therefore \csc{\theta} = \dfrac{1}{\dfrac{2\sqrt{2}}{3}}=\dfrac{3\sqrt{2}}{4}$
$\sec{\theta} = \dfrac{1}{\cos{\theta}}$
$\therefore \sec{\theta} = \dfrac{1}{-\dfrac{1}{3}}= -3$
$\cot{\theta} = \dfrac{1}{\tan{\theta}}$
$\cot{\theta} = \dfrac{1}{-2 \sqrt{2}}= -\dfrac{\sqrt{2}}{4}$
