1. **Problem Statement:** Given a right triangle with legs 14 and 50, find the trigonometric ratios for angle R. 2. **Recall the definitions:** - $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Find the hypotenuse:** $$\text{hypotenuse} = \sqrt{14^2 + 50^2} = \sqrt{196 + 2500} = \sqrt{2696}$$ 4. **Simplify the hypotenuse:** $$\sqrt{2696} = \sqrt{4 \times 674} = 2\sqrt{674} \approx 51.92$$ 5. **Given:** - $\sin Q = \frac{17}{50}$ (already provided) - $\cos Q = \frac{14}{50}$ (already provided) 6. **Find trigonometric ratios for angle R:** - Opposite side to R is 14 (adjacent to Q) - Adjacent side to R is 50 (opposite to Q) 7. **Calculate $\sin R$:** $$\sin R = \frac{\text{opposite to R}}{\text{hypotenuse}} = \frac{14}{51.92} \approx 0.2697 = \frac{14}{\sqrt{2696}}$$ 8. **Calculate $\cos R$:** $$\cos R = \frac{\text{adjacent to R}}{\text{hypotenuse}} = \frac{50}{51.92} \approx 0.9633 = \frac{50}{\sqrt{2696}}$$ 9. **Calculate $\tan R$:** $$\tan R = \frac{\text{opposite to R}}{\text{adjacent to R}} = \frac{14}{50} = \frac{7}{25}$$ **Final answers:** - $\sin R = \frac{14}{\sqrt{2696}} \approx 0.2697$ - $\cos R = \frac{50}{\sqrt{2696}} \approx 0.9633$ - $\tan R = \frac{7}{25}$