1. **State the problem:** Given $\sin \theta$ and $\cos \theta$, find the exact values of $\tan \theta$, $\csc \theta$, $\sec \theta$, and $\cot \theta$. 2. **Recall the definitions:** - $\tan \theta = \frac{\sin \theta}{\cos \theta}$ - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ 3. **Problem 35:** $\sin \theta = -\frac{3}{5}$, $\cos \theta = \frac{4}{5}$ Calculate each function: - $\tan \theta = \frac{-\frac{3}{5}}{\frac{4}{5}} = \frac{-3}{5} \times \frac{5}{4} = -\frac{3}{4}$ - $\csc \theta = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$ - $\sec \theta = \frac{1}{\frac{4}{5}} = \frac{5}{4}$ - $\cot \theta = \frac{\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{5} \times \frac{5}{-3} = -\frac{4}{3}$ 4. **Problem 37:** $\sin \theta = \frac{2\sqrt{5}}{5}$, $\cos \theta = \frac{\sqrt{5}}{5}$ Calculate each function: - $\tan \theta = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = \frac{2\sqrt{5}}{5} \times \frac{5}{\sqrt{5}} = 2$ - $\csc \theta = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{5}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{2 \times 5} = \frac{\sqrt{5}}{2}$ - $\sec \theta = \frac{1}{\frac{\sqrt{5}}{5}} = \frac{5}{\sqrt{5}} = \frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5}$ - $\cot \theta = \frac{\frac{\sqrt{5}}{5}}{\frac{2\sqrt{5}}{5}} = \frac{\sqrt{5}}{5} \times \frac{5}{2\sqrt{5}} = \frac{1}{2}$ **Final answers:** - For problem 35: $\tan \theta = -\frac{3}{4}$, $\csc \theta = -\frac{5}{3}$, $\sec \theta = \frac{5}{4}$, $\cot \theta = -\frac{4}{3}$ - For problem 37: $\tan \theta = 2$, $\csc \theta = \frac{\sqrt{5}}{2}$, $\sec \theta = \sqrt{5}$, $\cot \theta = \frac{1}{2}$