1. Stated problem: Calculate the sine of the largest angle in a triangle with sides 7, 8, and 9. 2. To find the sine of the largest angle, first identify the largest side, which is 9. 3. Use the Law of Cosines to find the largest angle $\theta$ opposite side 9: $$\cos(\theta) = \frac{7^2 + 8^2 - 9^2}{2 \cdot 7 \cdot 8} = \frac{49 + 64 - 81}{112} = \frac{32}{112} = \frac{2}{7}$$ 4. Calculate $\sin(\theta)$ using the identity $\sin^2(\theta) + \cos^2(\theta) = 1$: $$\sin(\theta) = \sqrt{1 - \cos^2(\theta)} = \sqrt{1 - \left(\frac{2}{7}\right)^2} = \sqrt{1 - \frac{4}{49}} = \sqrt{\frac{45}{49}} = \frac{3\sqrt{5}}{7}$$ 5. Final answer: The sine of the largest angle is $\boxed{\frac{3\sqrt{5}}{7}}$.