1. **Problem:** Evaluate the six trigonometric functions of the angle $\theta$ in a right triangle where the opposite side to $\theta$ is 9 and the hypotenuse is 15. 2. **Formula and rules:** The six trigonometric functions are defined as: - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\cot \theta = \frac{1}{\tan \theta}$ 3. **Find the adjacent side:** Using the Pythagorean theorem: $$\text{adjacent} = \sqrt{\text{hypotenuse}^2 - \text{opposite}^2} = \sqrt{15^2 - 9^2} = \sqrt{225 - 81} = \sqrt{144} = 12$$ 4. **Calculate each function:** - $\sin \theta = \frac{9}{15} = \frac{3}{5}$ - $\cos \theta = \frac{12}{15} = \frac{4}{5}$ - $\tan \theta = \frac{9}{12} = \frac{3}{4}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{3/5} = \frac{5}{3}$ - $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{4/5} = \frac{5}{4}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{1}{3/4} = \frac{4}{3}$ **Final answer:** $$\sin \theta = \frac{3}{5}, \quad \cos \theta = \frac{4}{5}, \quad \tan \theta = \frac{3}{4}, \quad \csc \theta = \frac{5}{3}, \quad \sec \theta = \frac{5}{4}, \quad \cot \theta = \frac{4}{3}$$