1. **State the problem:** We need to find the exact values of $\cot \theta$, $\sin \theta$, and $\csc \theta$ for the angle $\theta$ in a right triangle where the legs adjacent to $\theta$ are 3 (adjacent side) and 8 (opposite side). 2. **Recall the definitions:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$ - $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$ 3. **Find the hypotenuse:** Using the Pythagorean theorem: $$ \text{hypotenuse} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} $$ 4. **Calculate each value:** $$ \cot \theta = \frac{3}{8} $$ $$ \sin \theta = \frac{8}{\sqrt{73}} $$ $$ \csc \theta = \frac{\sqrt{73}}{8} $$ 5. **Final answers:** - $\cot \theta = \frac{3}{8}$ - $\sin \theta = \frac{8}{\sqrt{73}}$ - $\csc \theta = \frac{\sqrt{73}}{8}$