1. **State the problem:** We need to find the tangent of angle $S$ in a right triangle with sides $TS=10$ (opposite to $S$), $TU=24$ (adjacent to $S$), and hypotenuse $SU=26$. 2. **Recall the formula:** The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side: $$\tan(S) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Apply the formula:** Here, opposite side to $S$ is $TS=10$ and adjacent side to $S$ is $TU=24$. $$\tan(S) = \frac{10}{24}$$ 4. **Simplify the fraction:** $$\tan(S) = \frac{\cancel{10}}{\cancel{24}} = \frac{5}{12}$$ 5. **Final answer:** $$\boxed{\tan(S) = \frac{5}{12}}$$ This means the tangent of angle $S$ is $\frac{5}{12}$, which is the ratio of the opposite side to the adjacent side in the triangle.