Word Count : 471

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We are given two complex numbers: $$ z_1 = \frac{1}{2} - i \quad \text{and} \quad z_2 = 3 + i $$ We are to express $z_1$, $z_2$, and $z_1 z_2$ in: * Polar form: $r(\cos \theta + i \sin \theta)$ * Exponential form: $re^{i\theta}$ --- ### ✅ Step 1: Convert $z_1 = \frac{1}{2} - i$ #### 1.1 Modulus of $z_1$: $$ |z_1| = \sqrt{\left(\frac{1}{2}\right)^2 + (-1)^2} = \sqrt{\frac{1}{4} + 1} = \sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2} $$ #### 1.2 Argument of $z_1$ (denoted $\theta_1$): $$ \tan \theta_1 = \frac{-1}{1/2} = -2 $$ $$ \Rightarrow \theta_1 = __________ ______ ______ _______ ___ ___ ______.
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