1. **State the problem:** We have a right triangle with one leg of length 38, hypotenuse of length 58, and we need to find the missing angle $\theta$ opposite the leg of length 38. 2. **Formula used:** In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ 3. **Apply the formula:** $$\sin(\theta) = \frac{38}{58}$$ 4. **Simplify the fraction:** $$\sin(\theta) = \frac{\cancel{38}}{\cancel{58}} = \frac{19}{29}$$ 5. **Calculate the angle:** Use the inverse sine function: $$\theta = \sin^{-1}\left(\frac{19}{29}\right)$$ 6. **Evaluate the angle:** $$\theta \approx \sin^{-1}(0.6552) \approx 40.9^\circ$$ **Final answer:** The missing angle $\theta$ is approximately $40.9^\circ$.