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.

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

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

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

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

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

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

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

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

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