Okay, here we have this:
Considering the provided angle, we are going to calculate the requested trigonometric functions, so we obtain the following:
So the first thing we will do is calculate the length of the hypotenuse, that is, the distance between the given point and the origin, then we have:
[tex]\begin{gathered} r=\sqrt{(12-0)^2+(-5-0)^2} \\ r=\sqrt{12^2+(-5)^2} \\ r=\sqrt{144+25} \\ r=\sqrt{169} \\ r=13 \end{gathered}[/tex]Now we proceed to find the value of each ratio:
[tex]sin\beta=\frac{y}{r}=\frac{-5}{13}[/tex][tex]cos\beta=\frac{x}{r}=\frac{12}{13}[/tex][tex]tan\beta=\frac{y}{x}=\frac{-5}{12}[/tex][tex]csc\beta=\frac{r}{y}=\frac{13}{-5}=-\frac{13}{5}[/tex][tex]sec\beta=\frac{r}{x}=\frac{13}{12}[/tex][tex]cot\beta=\frac{x}{y}=\frac{12}{-5}=-\frac{12}{5}[/tex]