# How do you find the five remaining trigonometric function satisfying tantheta=5/4, costheta<0?

Jan 16, 2017

Find values of trig functions

#### Explanation:

Use trig identity:
$1 + {\tan}^{2} x = \frac{1}{{\cos}^{2} x}$
${\cos}^{2} x = \frac{1}{1 + {\tan}^{2} x}$
${\sin}^{2} x = \frac{1}{1 + {\cot}^{2} x}$
In this case:
${\cos}^{2} x = \frac{1}{1 + \frac{25}{16}} = \frac{1}{\frac{41}{16}} = \frac{16}{41}$
$\cos x = - \frac{4}{\sqrt{41}} = - \frac{4 \sqrt{41}}{41}$
Find sin x by the same way:
${\sin}^{2} x = \frac{1}{1 + \frac{16}{25}} = \frac{1}{\frac{41}{25}} = \frac{25}{41}$
$\sin x = - \frac{5}{\sqrt{41}}$ (because $\tan x = \frac{5}{4}$ > 0)
$\tan x = \frac{\sin}{\cos} = \frac{5}{4}$
$\cot x = \frac{1}{\tan} = \frac{4}{5}$
$\sec x = \frac{1}{\cos} = - \frac{\sqrt{41}}{4}$
$\csc x = \frac{1}{\sin} = - \frac{\sqrt{41}}{5}$