1. **Problem:** Find the indicated trigonometric ratios for the given right triangles. 2. **Recall SOH CAH TOA:** - $\sin = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Triangle XYZ (right angle at Z):** - Opposite side to $Z$ is hypotenuse $XY$ (unknown), legs $XZ=32$, $YZ=24$. - Calculate hypotenuse $XY$ using Pythagoras: $$XY=\sqrt{32^2+24^2}=\sqrt{1024+576}=\sqrt{1600}=40$$ 4. **Triangle ABC (right angle at B):** - Legs $AB=18$, $BC=30$, hypotenuse $CA$ unknown. - Calculate hypotenuse $CA$: $$CA=\sqrt{18^2+30^2}=\sqrt{324+900}=\sqrt{1224}=2\sqrt{306}$$ 5. **Calculate each ratio:** 1. $\sin A = \frac{\text{opposite to } A}{\text{hypotenuse}} = \frac{BC}{CA} = \frac{30}{2\sqrt{306}}$ 2. $\tan C = \frac{\text{opposite to } C}{\text{adjacent}} = \frac{AB}{BC} = \frac{18}{30} = \frac{3}{5}$ 3. $\cos X = \frac{\text{adjacent to } X}{\text{hypotenuse}} = \frac{YZ}{XY} = \frac{24}{40} = \frac{3}{5}$ 4. $\sin Z = \frac{\text{opposite to } Z}{\text{hypotenuse}} = \frac{XY}{XY} = 1$ 5. $\tan Z = \frac{\text{opposite to } Z}{\text{adjacent}} = \frac{XZ}{YZ} = \frac{32}{24} = \frac{4}{3}$ 6. $\cos A = \frac{\text{adjacent to } A}{\text{hypotenuse}} = \frac{AB}{CA} = \frac{18}{2\sqrt{306}}$ 7. $\sin X = \frac{\text{opposite to } X}{\text{hypotenuse}} = \frac{YZ}{XY} = \frac{24}{40} = \frac{3}{5}$ 8. $\cos C = \frac{\text{adjacent to } C}{\text{hypotenuse}} = \frac{BC}{CA} = \frac{30}{2\sqrt{306}}$ 9. $\tan A = \frac{\text{opposite to } A}{\text{adjacent}} = \frac{BC}{AB} = \frac{30}{18} = \frac{5}{3}$ 10. $\tan X = \frac{\text{opposite to } X}{\text{adjacent}} = \frac{XZ}{YZ} = \frac{32}{24} = \frac{4}{3}$ 6. **Simplify fractions where possible:** - $\sin A = \frac{30}{2\sqrt{306}} = \frac{15}{\sqrt{306}}$ - $\cos A = \frac{18}{2\sqrt{306}} = \frac{9}{\sqrt{306}}$ - $\cos C = \frac{30}{2\sqrt{306}} = \frac{15}{\sqrt{306}}$ **Final answers:** 1. $\sin A = \frac{15}{\sqrt{306}}$ 2. $\tan C = \frac{3}{5}$ 3. $\cos X = \frac{3}{5}$ 4. $\sin Z = 1$ 5. $\tan Z = \frac{4}{3}$ 6. $\cos A = \frac{9}{\sqrt{306}}$ 7. $\sin X = \frac{3}{5}$ 8. $\cos C = \frac{15}{\sqrt{306}}$ 9. $\tan A = \frac{5}{3}$ 10. $\tan X = \frac{4}{3}$