Finding functions for an angle whose terminal side passes through x,y

Question

How would I "Find the six trigonometric functions for the angle theta whose terminal side passes through the point (-8,-5)"?. I learned this material over 2 years ago and since then have forgotten. I don't even know where to start.

Answer 1

We would use the definitions for the trig functions in terms of $x$,$y$, and $r$.

In your example $$x=-8, \quad y=-5, \quad r=\sqrt{x^2+y^2} = \sqrt{(-5)^2 + (-8)^2} = \sqrt{89}$$

$$ \sin(\theta) = \frac{y}{r} = -\frac{5}{\sqrt{89}} $$

$$ \cos(\theta) = \frac{x}{r} = -\frac{8}{\sqrt{89}} $$

$$ \tan(\theta) = \frac{y}{x} = \frac{-5}{-8} = \frac{5}{8} $$

$$ \csc(\theta) = \frac{r}{y} = -\frac{\sqrt{89}}{5} $$

$$ \sec(\theta) = \frac{r}{x} = -\frac{\sqrt{89}}{8} $$

$$ \cot(\theta) = \frac{x}{y} = \frac{-8}{-5} = \frac{8}{5} $$