Complex Number Question: Write the given number in the polar form reiθ, -−π<θ≤π.

Question

Write the given number in the polar form re^iθ, -−π<θ≤π.

(6 - 6i)/(-sqrt(3) + i)

Answer 1

(6 - 6i) / (-√3 + i)

6 - 6i = 6(1 - i) = (6√2)(1/√2 - (1/√2) i) = 6√2[cos(7π/4) + i sin(7π/4)]

-√3 + i = 2(-√3/2 + (1/2) i) = 2[cos(5π/6) + i sin(5π/6)]

Using DeMoivre's Theorem,

(6 - 6i) / (-√3 + i) =

[6√2 / 2] [ cos(7π/4 - 5π/6) + i sin(7π/4 - 5π/6) ] = 3√2[cos(11π/12) + isin(11π/12)] = (3√2)e^{i (11π/12)}

Answer 2

multiply by 1 to clear denominator

z=[(6-6i)/(-√3+i)}*[(-√3-i)/(-√3-i)]

=[(-6√3-6)+i(6√3-6)]/4

note the point is in second quadrant since real part negative and i part positive.

θ=π+tan^-1[(6√3-6)/(-6√3-6)]

θ=2.87 radian

r=[([(-6√3-6)²+(-6√3+6)²)]^1/2]/4

r=16.97/4=4.24