Answer to Question #344843 in Complex Analysis for Dkay

Question

given z₁ = 2<45 degrees, z₂ =3<120 degrees and z₃ =4<180 degrees. determine the following and leave your answer in rectangular form.

1. (z₁)²+z₂/z₂+z₃
2. z_1/z₂z₃

Answer 1

1.

$(z_1)^2=(2)^2\angle(2\cdot45\degree)=4\angle90\degree=4i$

$z_2+z_3=3\angle120\degree+4\angle180\degree$

$=3(-\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}i)+4(-1)$

$=-\dfrac{11}{2}+\dfrac{3\sqrt{3}}{2}i$

$\dfrac{z_2}{z_2+z_3}=\dfrac{-\dfrac{3}{2}+\dfrac{3\sqrt{3}}{2}i}{-\dfrac{11}{2}+\dfrac{3\sqrt{3}}{2}i}=\dfrac{3-3\sqrt{3}i}{11-3\sqrt{3}i}$

$=\dfrac{(3-3\sqrt{3}i)(11+3\sqrt{3}i)}{121+27}$

$=\dfrac{33+9\sqrt{3}i-33\sqrt{3}i+27}{148}$

$=\dfrac{15}{37}-\dfrac{6}{37}i$

$(z_1)^2+\dfrac{z_2}{z_2+z_3}=4i+\dfrac{15}{37}-\dfrac{6}{37}i$

$=\dfrac{15}{37}+\dfrac{142}{37}i$

2.

$z_2z_3=3\angle120\degree(4\angle180\degree)$ $=3(4)\angle(120\degree+180\degree)=12\angle300\degree$

$=12(\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i)=6-6\sqrt{3}i$

$\dfrac{z_1}{z_2z_3}=\dfrac{2}{12}\angle(45\degree-300\degree)$

$=\dfrac{1}{6}(\cos(-255\degree)+i\sin(-255\degree))$

$=\dfrac{1}{6}(\cos(105\degree)+i\sin(105\degree))$

$\dfrac{z_1}{z_2z_3}=\dfrac{\sqrt{2}+\sqrt{2}i}{6-6\sqrt{3}i}$

$=\dfrac{(\sqrt{2}+\sqrt{2}i)(6+6\sqrt{3}i)}{36+108}$

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

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