Answer
$ \sin{t} = -\dfrac{2}{3}$
$ \cos{t} = -\dfrac{\sqrt{5}}{3}$
$\tan{t} =\dfrac{2\sqrt{5}}{5}$
$\csc{t} =-\dfrac{3}{2}$
$\sec{t} =-\dfrac{3\sqrt{5}}{5}$
$\cot{t} =\dfrac{\sqrt{5}}{2}$
Work Step by Step
With $P= \left(-\dfrac{\sqrt{5}}{3},-\dfrac{2}{3} \right) = (x,y)$, then $x = -\dfrac{\sqrt{5}}{3} \text{ and } \hspace{10pt} y =-\dfrac{2}{3}$
Thus,
$\sin{t} = y$
$ \sin{t} = -\dfrac{2}{3}$
$\cos{t} = x$
$ \cos{t} = -\dfrac{\sqrt{5}}{3}$
$\tan{t} = \dfrac{y}{x}$
$\tan{t} = \dfrac{-\dfrac{2}{3}}{ -\dfrac{\sqrt{5}}{3} } = \dfrac{2\sqrt{5}}{5}$
$\csc{t} = \dfrac{1}{y}$
$\csc{t} = \dfrac{1}{-\dfrac{2}{3}} = -\dfrac{3}{2}$
$\sec{t} = \dfrac{1}{x}$
$\sec{t} = \dfrac{1}{-\dfrac{\sqrt{5}}{3}}= -\dfrac{3\sqrt{5}}{5}$
$\cot{t} = \dfrac{x}{y}$
$\cot{t} = \dfrac{-\dfrac{\sqrt{5}}{3} }{-\dfrac{2}{3}} = \dfrac{\sqrt{5}}{2}$
