1. **State the problem:** Find the exact values of $\sin \theta$, $\cot \theta$, and $\csc \theta$ for the given right triangle with legs 8 (base) and 15 (vertical), and angle $\theta$ at the bottom-left. 2. **Use the Pythagorean theorem to find the hypotenuse:** $$ \text{hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 $$ 3. **Recall the definitions of the trigonometric ratios:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$ 4. **Identify the sides relative to $\theta$:** - Opposite side = 15 - Adjacent side = 8 - Hypotenuse = 17 5. **Calculate each ratio:** $$ \sin \theta = \frac{15}{17} $$ $$ \cot \theta = \frac{8}{15} $$ $$ \csc \theta = \frac{17}{15} $$ **Final answers:** - $\sin \theta = \frac{15}{17}$ - $\cot \theta = \frac{8}{15}$ - $\csc \theta = \frac{17}{15}$