Answer
See below
Work Step by Step
Substitute $(5,-12)$ into $x^2+y^2=r^2$ to find $r$:
$$x^2+y^2=r^2\\5^2+(-12)^2=r^2\\r^2=169\\r=\pm 13\\r=13$$
$\sin \theta=\frac{y}{r}=-\frac{12}{13}$
$\cos \theta=\frac{x}{r}=\frac{5}{13}$
$\csc \theta=\frac{r}{y}=-\frac{13}{12}$
$\sec \theta=\frac{r}{x}=\frac{13}{5}$
$\tan \theta=\frac{y}{x}=-\frac{12}{5}$
$\cot \theta=\frac{x}{y}=-\frac{5}{12}$
