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