The point (8,-15) 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?

Aug 16, 2017

Explanation:

Memorize this

$\left(x , y\right)$ lies on the terminal side of $\theta$, then

$r = \sqrt{{x}^{2} + {y}^{2}}$ and

$\sin \theta = \frac{y}{r}$ $\text{ }$ $\text{ }$ $\text{ }$ $\csc \theta = \frac{r}{y}$

$\cos \theta = \frac{x}{r}$ $\text{ }$ $\text{ }$ $\text{ }$ $\sec \theta = \frac{r}{x}$

$\tan \theta = \frac{y}{x}$ $\text{ }$ $\text{ }$ $\text{ }$ $\cot \theta = \frac{x}{y}$

For this question

We have $\left(x , y\right) = \left(8 , - 15\right)$
Do the arithmetic and substitute.

$r = \sqrt{{\left(8\right)}^{2} + {\left(- 15\right)}^{2}} = 17$

So

$\sin \theta = - \frac{15}{17}$ $\text{ }$ $\text{ }$ $\text{ }$ $\csc \theta = - \frac{17}{15}$

$\cos \theta = \frac{8}{17}$ $\text{ }$ $\text{ }$ $\text{ }$ $\sec \theta = \frac{17}{8}$

$\tan \theta = - \frac{15}{8}$ $\text{ }$ $\text{ }$ $\text{ }$ $\cot \theta = - \frac{8}{15}$