1. **State the problem:** Find the exact values of the six trigonometric functions for a right triangle with sides 9, 40, and hypotenuse 41, angle $\theta$ opposite side 40. 2. **Recall the definitions:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ - Reciprocals: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, $\cot \theta = \frac{1}{\tan \theta}$ 3. **Apply values:** - $\sin \theta = \frac{40}{41}$ - $\cos \theta = \frac{9}{41}$ - $\tan \theta = \frac{40}{9}$ 4. **Find reciprocals:** - $\csc \theta = \frac{41}{40}$ - $\sec \theta = \frac{41}{9}$ - $\cot \theta = \frac{9}{40}$ 5. **Interpretation:** These ratios come directly from the triangle sides relative to angle $\theta$. This method applies to any right triangle. This is the process to solve the first problem. For other problems, use the same approach: identify sides, apply definitions, use Pythagorean theorem if needed, and calculate ratios.