# How do you find the other five trigonometric functions of x if cosx = 3/5?

Aug 8, 2015

It;s crucial to be familiar with the Pythagorean trigonometric identity and trigonometric identities corresponding to relations of sine and cosine with other trigonometric functions.

#### Explanation:

First we use the Pythagorean trigonometric identity:
${\sin}^{2} x + {\cos}^{2} x = 1$
${\sin}^{2} x + {\left(\frac{3}{5}\right)}^{2} = 1$
${\sin}^{2} x + \frac{9}{25} = 1$
${\sin}^{2} x = 1 - \frac{9}{25}$
${\sin}^{2} x = \frac{16}{25}$
$\sin x = \pm \frac{4}{5}$
Now, if we knew more about the angle $x$ we could eliminate one of the possibilities for $\sin x$ but since we don't we should examine both options:

CASE 1: $\sin x = \frac{4}{5}$
Using basic trigonometric identities:
$\tan x = \sin \frac{x}{\cos} x = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{5} \cdot \frac{5}{3} = \frac{4}{3}$
$\cot x = \cos \frac{x}{\sin} x = \frac{3}{4}$
$\sec x = \frac{1}{\cos} x = \frac{5}{3}$
$\csc x = \frac{1}{\sin} x = \frac{5}{4}$

CASE 2: $\sin x = - \frac{4}{5}$
Using basic trigonometric identities:
$\tan x = \sin \frac{x}{\cos} x = \frac{- \frac{4}{5}}{\frac{3}{5}} = - \frac{4}{5} \cdot \frac{5}{3} = - \frac{4}{3}$
$\cot x = \cos \frac{x}{\sin} x = - \frac{3}{4}$
$\sec x = \frac{1}{\cos} x = \frac{5}{3}$
$\csc x = \frac{1}{\sin} x = - \frac{5}{4}$