The point (-5,-2) is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

Jan 27, 2018

$\sin \theta \approx - 0.37$
$\cos \theta \approx - 0.93$
$\tan \theta \approx 0.40$
$\csc \theta \approx - 2.69$
$\sec \theta \approx - 1.08$
$\cot \theta \approx 2.50$

Explanation:

Here's a diagram I made using Desmos: The unknown angle $\theta$ can be calculated by using $\tan$ of the lengths we already know:

theta = 180º + tan^-1(2/5)

~~180º+21.8º=201.8º

You can use a calculator or a sine table to calculate the trig functions:

$\sin \theta \approx - 0.37$
$\cos \theta \approx - 0.93$
$\tan \theta \approx 0.40$
$\csc \theta \approx - 2.69$
$\sec \theta \approx - 1.08$
$\cot \theta \approx 2.50$

If your calculator doesn't support $\tan$, $\csc$, $\sec$, or $\cot$, here are some helpful conversions you can use:

$\tan \theta = \sin \frac{\theta}{\cos} \theta$

$\csc \theta = \frac{1}{\sin} \theta$

$\sec \theta = \frac{1}{\cos} \theta$

$\cot \theta = \frac{1}{\tan} \theta = \frac{1}{\sin \frac{\theta}{\cos} \theta} = \cos \frac{\theta}{\sin} \theta$