1. State the problem (Question 17): Given the terminal point for angle $\theta$ is $(-4,-7)$, find the values of the six trigonometric functions: $\sin\theta,\cos\theta,\tan\theta,\csc\theta,\sec\theta,\cot\theta$. 2. Use the key triangle/coordinate relationships: Let the terminal point be $(x,y)$ and $r=\sqrt{x^2+y^2}$. Then: $\sin\theta=\frac{y}{r}$ $\cos\theta=\frac{x}{r}$ $\tan\theta=\frac{y}{x}$ $\csc\theta=\frac{r}{y}$ $\sec\theta=\frac{r}{x}$ $\cot\theta=\frac{x}{y}$ 3. Identify $x$, $y$, and compute $r$: $x=-4$ y=-7$ $$r=\sqrt{(-4)^2+(-7)^2}=\sqrt{16+49}=\sqrt{65}$$ 4. Compute $\sin\theta$: $\sin\theta=\frac{y}{r}=\frac{-7}{\sqrt{65}}$ 5. Compute $\cos\theta$: $\cos\theta=\frac{x}{r}=\frac{-4}{\sqrt{65}}$ 6. Compute $\tan\theta$: $\tan\theta=\frac{y}{x}=\frac{-7}{-4}$ $\tan\theta=\cancel{(-7)}\div\cancel{(-4)}=\frac{7}{4}$ 7. Compute $\csc\theta$: $\csc\theta=\frac{r}{y}=\frac{\sqrt{65}}{-7}$ 8. Compute $\sec\theta$: $\sec\theta=\frac{r}{x}=\frac{\sqrt{65}}{-4}$ 9. Compute $\cot\theta$: $\cot\theta=\frac{x}{y}=\frac{-4}{-7}$ $\cot\theta=\cancel{(-4)}\div\cancel{(-7)}=\frac{4}{7}$ 10. Final answers for Question 17: $\sin\theta=\frac{-7}{\sqrt{65}}$ $\cos\theta=\frac{-4}{\sqrt{65}}$ $\tan\theta=\frac{7}{4}$ $\csc\theta=\frac{-\sqrt{65}}{7}$ $\sec\theta=\frac{-\sqrt{65}}{4}$ $\cot\theta=\frac{4}{7}$