# How do you use the point (3,6) on the terminal side of the angle to evaluate the six trigonometric functions?

Jun 27, 2017

$\tan x = \frac{y}{x} = \frac{6}{3} = 2$
${\cos}^{2} x = \frac{1}{1 + {\tan}^{2} x} = \frac{1}{1 + 4} = \frac{1}{5}$
$\cos x = \pm \frac{1}{\sqrt{5}}$
${\sin}^{2} x = 1 - {\cos}^{2} x = 1 - \frac{1}{5} = \frac{4}{5}$
$\sin x = \pm \frac{2}{\sqrt{5}} = \pm \frac{2 \sqrt{5}}{5}$
$\cot x = \frac{1}{\tan} = \frac{1}{2}$
$\sec x = \frac{1}{\cos} = \pm \sqrt{5}$
$\csc x = \frac{1}{\sin} = \pm \frac{\sqrt{2}}{5}$

Note. The reason we have 2 opposite values for sin x, and cos x, is
because there are 2 arcs (angles) that have the same tan value.
$x = {63}^{\circ} 43$ and $x = 63.43 + 180 = {243}^{\circ} 43$