1. **State the problem:** We need to find the tangent of angle $S$ in a right triangle $TSR$ where the right angle is at $T$. 2. **Recall the definition of tangent:** For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. 3. **Identify the sides relative to angle $S$:** - Opposite side to $S$ is $12$. - Adjacent side to $S$ is $35$. 4. **Apply the formula:** $$\tan(S) = \frac{12}{35}$$ 5. **Simplify the fraction if possible:** - $12$ and $35$ have no common factors other than $1$, so the fraction is already in simplest form. 6. **Final answer:** $$\boxed{\tan(S) = \frac{12}{35}}$$