1. **Problem statement:** Find $\cos(\beta)$ in the right triangle with sides $BC=5$, $AC=12$, and hypotenuse $AB=13$, where $\beta$ is the angle at vertex $B$. 2. **Formula:** In a right triangle, $\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$. 3. **Identify sides relative to angle $\beta$:** - The side adjacent to angle $\beta$ is $AC = 12$. - The hypotenuse is $AB = 13$. 4. **Calculate $\cos(\beta)$:** $$\cos(\beta) = \frac{AC}{AB} = \frac{12}{13}$$ 5. **Conclusion:** The value of $\cos(\beta)$ is $\frac{12}{13}$, which corresponds to answer choice B.