Answer to Question #348766 in Complex Analysis for Busi

Question

Write z1=(√3+i) and z2=(1−i) in trigonometric form. Then the argument of the quotient (z1)/(z2)is given by

Answer 1

$$r^2=x^2+y^2=(\sqrt{3})^2+(1)^2=4$$ $$\tan\theta=\dfrac{y}{x}=\dfrac{1}{\sqrt{3}}$$

Quadrant I

$$z_1=\sqrt{3}+i=2(\cos(\dfrac{\pi}{6})+i\sin(\dfrac{\pi}{6}))$$

$$r^2=x^2+y^2=(1)^2+(-1)^2=2$$ $$\tan\theta=\dfrac{-1}{1}=-1$$

Quadrant IV

$$z_2=1-i=\sqrt{2}(\cos(\dfrac{7\pi}{4})+i\sin(\dfrac{7\pi}{4}))$$

$z_1/z_2$

$$\theta_1-\theta_2=\dfrac{\pi}{6}-\dfrac{7\pi}{4}=\dfrac{-19\pi}{12}$$

$$Arg(z_1/z_2)=\dfrac{-19\pi}{12}+2\pi=\dfrac{5\pi}{12}$$