1. **State the problem:** We need to find the value of $\sin Z$ in the right triangle with vertices $Y$, $Z$, and $X$, where the right angle is at $Y$. 2. **Identify the sides:** The side opposite the right angle $Y$ is the hypotenuse, which is the side $ZX$ with length 20. The side opposite angle $Z$ is $YX$ with length $\sqrt{43}$. 3. **Recall the sine definition:** For any angle in a right triangle, $\sin(\text{angle}) = \frac{\text{opposite side}}{\text{hypotenuse}}$. 4. **Apply the formula for $\sin Z$:** $$\sin Z = \frac{\text{opposite side to } Z}{\text{hypotenuse}} = \frac{YX}{ZX} = \frac{\sqrt{43}}{20}$$ 5. **Calculate the decimal value:** $$\sqrt{43} \approx 6.5574$$ $$\sin Z \approx \frac{6.5574}{20} = 0.32787$$ 6. **Round to the nearest hundredth:** $$\sin Z \approx 0.33$$ **Final answer:** $\sin Z = 0.33$