1. **State the problem:** We need to find $\cot \theta$, $\cos \theta$, and $\sec \theta$ for the angle $\theta$ in a right triangle where the hypotenuse is 10, the vertical leg (opposite side) is 9, and the base (adjacent side) is 4. 2. **Recall the definitions:** - $\cot \theta = \frac{\text{adjacent}}{\text{opposite}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$ 3. **Identify sides relative to $\theta$:** - Opposite side = 9 - Adjacent side = 4 - Hypotenuse = 10 4. **Calculate $\cot \theta$:** $$ \cot \theta = \frac{4}{9} $$ 5. **Calculate $\cos \theta$:** $$ \cos \theta = \frac{4}{10} = \frac{2}{5} $$ 6. **Calculate $\sec \theta$:** $$ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{2}{5}} = \frac{5}{2} $$ **Final answers:** - $\cot \theta = \frac{4}{9}$ - $\cos \theta = \frac{2}{5}$ - $\sec \theta = \frac{5}{2}$