1. **Problem statement:** Given that $\cos \theta = \frac{1}{2}$ and $\sin \theta = \frac{\sqrt{3}}{2}$, find the measure of angle $\theta$. 2. **Recall the unit circle values:** The cosine and sine values correspond to specific angles on the unit circle. Important angles to remember are: - $\cos \frac{\pi}{3} = \frac{1}{2}$ and $\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ - $\cos \frac{5\pi}{6} = -\frac{\sqrt{3}}{2}$ and $\sin \frac{5\pi}{6} = \frac{1}{2}$ (not matching our values) 3. **Check quadrant:** Since $\cos \theta > 0$ and $\sin \theta > 0$, $\theta$ lies in the first quadrant where both sine and cosine are positive. 4. **Match values:** The values $\cos \theta = \frac{1}{2}$ and $\sin \theta = \frac{\sqrt{3}}{2}$ exactly match the angle $\theta = \frac{\pi}{3}$. 5. **Conclusion:** The measure of angle $\theta$ is $\boxed{\frac{\pi}{3}}$. **Answer:** (a) $\frac{\pi}{3}$