How do you find all six trigonometric function of theta if the point (-3,sqrt7) is on the terminal side of theta?

Apr 6, 2017

$\tan t = - \frac{\sqrt{7}}{3}$, and t is in Quadrant 2.
Use trig identity:
${\cos}^{2} t = \frac{1}{1 + {\tan}^{2} t} = \frac{1}{1 + \frac{7}{9}} = \frac{9}{16}$
$\cos t = \pm \frac{3}{4}$
Since t is in Quadrant 2, then, cos t is negative.
$\cos t = - \frac{3}{4}$
$\sin t = \tan t . \cos t = \left(- \frac{\sqrt{7}}{3}\right) \left(- \frac{3}{4}\right) = \frac{\sqrt{7}}{4}$
$\cot t = \frac{1}{\tan} = - \frac{3}{\sqrt{7}}$
$\sec t = \frac{1}{\cos} = - \frac{4}{3}$
$\csc t = \frac{1}{\sin} = \frac{4}{\sqrt{7}} = \frac{4 \sqrt{7}}{7}$