Write z1 and z2 in polar form z1 4 Squareroot 3 4i z2 1

Write z_1 and z_2 in polar form. z_1 = 4 Squareroot 3 - 4i, z_2 = -1 + I z_1 = ___________ z_2 = ____________ Find the product z_1z_2 and the quotients z_1/z_2 and 1/z_1. (Express your answers in polar form.) z_1z_2 = _________ z_1/z_2 = _________ 1/z_1 = _________

Solution

z1=4sqrt3-4i

r=sqrt(x^2+y^2)=sqrt((4sqrt3)^2 + 4^2)=8

tan theta= y/x= -4/4sqrt3=-1/sqrt3

theta=-30

therefore polar form is 8(cos(-30)+ isin(-30))=8(cos 30 -isin 30)

z2= -1+i

r=sqrt(x^2+y^2)=sqrt((-1)^2 + 1^2)=sqrt2

tan theta=y/x=1/-1

theta=-45

therefore required polar form is sqrt2(cos(-45 )+isin(-45))=sqrt2(cos 45-isin45)