# The terminal side of theta in standard position contains (5,12), how do you find the exact values of the six trigonometric functions of theta?

Apr 1, 2017

$\sin \theta = \frac{12}{13}$, $\cos \theta = \frac{5}{13}$, $\tan \theta = \frac{12}{5}$

$\csc \theta = \frac{13}{12}$, $\sec \theta = \frac{13}{5}$, $\cot \theta = \frac{5}{12}$

#### Explanation:

Let $\left(x , y\right) = \left(5 , 12\right)$.

$r = \sqrt{{x}^{2} + {y}^{2}} = \sqrt{{5}^{2} + {12}^{2}} = 13$

By definition,

$\sin \theta = \frac{y}{r} = \frac{12}{13}$

$\cos \theta = \frac{x}{r} = \frac{5}{13}$

$\tan \theta = \frac{y}{x} = \frac{12}{5}$

$\csc \theta = \frac{r}{y} = \frac{13}{12}$

$\sec \theta = \frac{r}{x} = \frac{13}{5}$

$\cot \theta = \frac{x}{y} = \frac{5}{12}$

I hope that this was clear.