Answer to Question #349118 in Complex Analysis for Busi

Question

(2<45°)/((3<120°)(4<180°)) leave your answer in rectangular form

Answer 1

$$z_1=2(\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}i)=\sqrt{2}+\sqrt{2}i$$

$$z_2=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)=-\dfrac{3}{2}+\dfrac{3\sqrt{3}}{2}i$$

$$z_3=-4i$$

$$z_2z_3=(-\dfrac{3}{2}+\dfrac{3\sqrt{3}}{2}i)(-4i)=6\sqrt{3}+6i$$

$$\dfrac{z_1}{z_2z_3}=\dfrac{\sqrt{2}(1+i)}{6(\sqrt{3}+i)}$$

$$=\dfrac{\sqrt{2}(1+i)(\sqrt{3}-i)}{24}$$

$$=\dfrac{\sqrt{6}-\sqrt{2}i+\sqrt{6}i+\sqrt{2}}{24}$$

$$=\dfrac{\sqrt{6}+\sqrt{2}}{24}+\dfrac{\sqrt{6}-\sqrt{2}}{24}i$$