# How do you find all six trigonometric function of theta if the point (3,4) is on the terminal side of theta?

Mar 20, 2017

The terminal side, containing point (3, 4) is located in Quadrant 1.
Call t the angle:
$\tan t = \frac{y}{x} = \frac{4}{3}$
${\cos}^{2} t = \frac{1}{1 + {\tan}^{2} t} = \frac{1}{1 + \frac{16}{9}} = \frac{9}{25}$
$\cos t = \frac{3}{5}$ (t in Quadrant 1 --> cos t is positive)
${\sin}^{2} t = 1 - {\cos}^{2} t = 1 - \frac{9}{25} = \frac{16}{25}$
$\sin t = \frac{4}{5}$ (t in Quadrant 1 --> sin t is positive)
$\tan t = \frac{4}{3}$
$\cot t = \frac{1}{\tan t} = \frac{3}{4}$
$\sec t = \frac{1}{\cos t} = \frac{5}{3}$
$\csc t = \frac{1}{\sin t} = \frac{5}{4}$