# How do you find all six trigonometric function of theta if the point (0,-5) is on the terminal side of theta?

Mar 3, 2017

$\sin \theta = 0 , \cos \theta = - 1$
csc theta = "undefined"; sec theta = -1
$\tan \theta = 0 , \cot \theta = \text{undefined}$

#### Explanation:

When you place the point $\left(0 , - 5\right)$ on a coordinate plane, the angle from the x-axis to the point is $\theta = {180}^{\circ}$counterclockwise.

$\sin {180}^{\circ} = 0$, which is the $y$ distance from the point to the $x$-axis.

$\cos {180}^{\circ} = - 1$ from a trig. circle.

So $r \cos \theta = - 1$

From these two trig. values we can determine the rest using the trig. function definitions:

$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{0} = \text{undefined}$

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

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{0}{-} 1 = 0$

$\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = - \frac{1}{0} = \text{undefined}$

In Summary:

$\sin \theta = 0 , \cos \theta = - 1$
csc theta = "undefined"; sec theta = -1
$\tan \theta = 0 , \cot \theta = \text{undefined}$