1. **Problem statement:** Calculate the value of $\theta$ in the first triangle where the opposite side to $\theta$ is 13 and the hypotenuse is 19. 2. **Formula:** Use the sine ratio for right triangles: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 3. **Apply values:** $$\sin(\theta) = \frac{13}{19}$$ 4. **Calculate $\theta$:** Use the inverse sine function: $$\theta = \sin^{-1}\left(\frac{13}{19}\right)$$ 5. **Evaluate:** $$\theta \approx \sin^{-1}(0.6842) \approx 43.16^\circ$$ 6. **Answer:** The value of $\theta$ to the nearest hundredth is **43.16 degrees**.