# The terminal side of theta in standard position contains (4,-2), how do you find the exact values of the six trigonometric functions of theta?

Feb 3, 2018

See explanation.

#### Explanation:

The point's coordinates are:

$x = 4$, $y = - 2$

To calculate the functions we have to calculate the distance between the point and the origin:

$r = \sqrt{{x}^{2} + {y}^{2}} = \sqrt{{4}^{2} + {\left(- 2\right)}^{2}} = \sqrt{16 + 4} = \sqrt{20} = 2 \sqrt{5}$

Now we can calculate the functions:

$\sin \theta = \frac{y}{r} = - \frac{2}{2 \sqrt{5}} = - \frac{\sqrt{5}}{5}$

$\cos \theta = \frac{x}{r} = \frac{4}{2 \sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}$

$\tan \theta = \frac{y}{x} = - \frac{2}{4} = - \frac{1}{2}$

$\cot \theta = \frac{x}{y} = \frac{4}{-} 2 = - 2$

$\sec \theta = \frac{r}{x} = \frac{2 \sqrt{5}}{4} = \frac{\sqrt{5}}{2}$

$\csc \theta = \frac{r}{y} = \frac{2 \sqrt{5}}{-} 2 = - \sqrt{5}$