1. **Stating the problem:** We are given a right triangle with vertices A, an unlabeled top vertex, and O on the circle's circumference. Side AO (horizontal) is 3 cm, and the vertical side from A to the top vertex is 4 cm. The angle at vertex O is $\theta$. We need to find the magnitude of angle $\theta$. 2. **Identify the sides relative to angle $\theta$:** - The side adjacent to angle $\theta$ (horizontal) is 3 cm (AO). - The side opposite angle $\theta$ (vertical) is 4 cm. 3. **Use the tangent function:** Since the triangle is right-angled, $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3}$$ 4. **Calculate $\theta$:** $$\theta = \tan^{-1}\left(\frac{4}{3}\right)$$ Using a calculator, $$\theta \approx 53.13^{\circ}$$ 5. **Conclusion:** The magnitude of angle $\theta$ is approximately $53.13^{\circ}$.