1. **State the problem:** Convert each complex number $z_k$ into the form $\cos\theta + i\sin\theta$ by finding the angle $\theta$. 2. **Recall the formula:** A complex number in rectangular form $a + bi$ can be written as $\cos\theta + i\sin\theta$ where $\theta = \arg(z) = \tan^{-1}(\frac{b}{a})$. 3. **Important rules:** - The angle $\theta$ must be adjusted based on the quadrant of $(a,b)$. - Use exact values for common angles. --- **1) For** $z_1 = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$: - $a = \frac{\sqrt{2}}{2}$, $b = \frac{\sqrt{2}}{2}$ - $\theta = \tan^{-1}(1) = \frac{\pi}{4}$ (1st quadrant) - So, $z_1 = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$ **2) For** $z_2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$: - $a = -\frac{1}{2}$, $b = -\frac{\sqrt{3}}{2}$ - $\tan\theta = \frac{b}{a} = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}$ - Reference angle $= \frac{\pi}{3}$ - Since $a<0$ and $b<0$, $z_2$ is in the third quadrant - $\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}$ - So, $z_2 = \cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3}$ **3) For** $z_3 = -\frac{\sqrt{3}}{2} + \frac{1}{2}i$: - $a = -\frac{\sqrt{3}}{2}$, $b = \frac{1}{2}$ - $\tan\theta = \frac{b}{a} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}$ - Reference angle $= \frac{\pi}{6}$ - Since $a<0$ and $b>0$, $z_3$ is in the second quadrant - $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$ - So, $z_3 = \cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6}$ **4) For** $z_4 = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}i$: - $a = -\frac{\sqrt{2}}{2}$, $b = -\frac{\sqrt{2}}{2}$ - $\tan\theta = \frac{b}{a} = 1$ - Reference angle $= \frac{\pi}{4}$ - Since $a<0$ and $b<0$, $z_4$ is in the third quadrant - $\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}$ - So, $z_4 = \cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4}$ **5) For** $z_5 = i$: - $a = 0$, $b = 1$ - This lies on the positive imaginary axis - $\theta = \frac{\pi}{2}$ - So, $z_5 = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2}$ --- **Final answers:** $$ z_1 = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4} $$ $$ z_2 = \cos\frac{4\pi}{3} + i\sin\frac{4\pi}{3} $$ $$ z_3 = \cos\frac{5\pi}{6} + i\sin\frac{5\pi}{6} $$ $$ z_4 = \cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4} $$ $$ z_5 = \cos\frac{\pi}{2} + i\sin\frac{\pi}{2} $$