\( |z| = \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{3}{2}\right)^2} = \) \( \sqrt{\frac{1}{4} + \frac{3}{4}} = \sqrt{1} = 1 \)

\( \theta = \arctan\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \) \( \arctan\frac{\sqrt{3}}{2}2 = \arctan\sqrt{3} = \frac{\pi}{3} \)