Question

Express z = 1 + 2j in polar form then find z^3 then convert the answer to Cartesian
coordinates.

Answer 1

I am used to expressing complex numbers in the form a + bi, instead of your a + bj

if z = 1 + 2i
r = √(1^2+2^2) = √5
and if Ø is our angle, tan Ø = 2/1 , Ø is in I of your complex plane
Ø = appr 1.107 radians

then z = √5(cos 1.107 + i sin 1.107) or √5 cis 1.107

z^3 = √5^3 cis (3(1.107)) using De Moivre's Theorem
= √5^3 cis 3.3214..
or
= √5^3 (cos 3.3214.. + i sin 3.3214..)
= (-11 + (-2) i)

z^3 = -11 - 2i

....

we could have done this:
(1+2i)^3
= (1+2i)(1 + 4i + 4i^2)
= (1+2i)(-3 + 4i)
= -3+4i - 6i + 8i^2
= -11 - 2i

Answer 2

Can the equation be expressed in a polar form diagram?