Write z1 and z2 in polar formz1 32 z2 sqrt 3 i z1 z2 Find

Write z₁ and z₂ in polar form.z₁ = 32, z₂ = sqrt 3 +i

z1=

z2=

Find the product z₁z₂ and the quotients z1 / z2 and 1 / z1. (Express your answers in polar form.)

z1z2=

z1/z2=

1/z1=

Thank you!!

Solution

z1 = -32 ; z2 = 3+i

z = x+ iy ; In polar for, z = re^i*theta ; r = sqrt(x^2 + y^2) ; theta = tan^-1(y/x)

z1 = 32e^i*pi

z2: r =sqrt(9 +1) = sqrt10 ; theta = tan^-1(1/3) = 1.249 rad

z2 = sqrt10e^i*1.249

So, z1.z2 = 32e^i*pi*sqrt10e^i*1.249 = 101.2e^i*4.39

z1/z2 = 32e^i*pi/sqrt10e^i*1.249 = 3.2e^i*1.89

1/z1 = 1/ 32e^i*pi = 0.03125e^-i*pi