Question

Write z1 and z2 in polar form. (Express θ in radians.)
z1 = [(2)^1/2] − [(2)^1/2] i, z2 = 1 − i

Answer 1

z1=\[\sqrt{2}-\sqrt{2}i\]

Answer 2

|z1| = sqrt(sqrt(2)^2 + sqrt(2)^2)
= sqrt(2+2)
= sqrt(4)
= 2
arg(z1) = arctan(-sqrt(2)/sqrt(2))
= arctan(-1)
= -pi/4 or 3pi/4
Re(z1) = +ve, Im(z1) = -ve, therefore -pi/2 < arg(z1) < 0
therefore
= -pi/4

Answer 3

|z2| = sqrt(1^2 + 1^2)
= sqrt(2)
arg(z2) = arctan(-1/1)
= arctan(-1)
and then follows the same logic as for z1
so the argument is also -pi/4

Answer 4

so overall:
z1 = 2 cis -pi/4
z2 = sqrt(2) cis -pi/4

Answer 5

this stuff really clicks for me now. awesome responses. thanks.