conver to polar form?

Question

can you convert thjis to polar form

-3sqrt(7)-sqrt(21)i

Answer 1

Let z = a +bi = -3√7 -√21 i = √21 ( -√3 - i )
(factoring out a common √21 ...)

Magnitude |z| = √(a² + b²) = √21 √((-√3)² + (-1)²) = 2√21
Angle z = ∠z = arctan(b/a) = arctan((-1)/(-√3)) = arctan(1/√3)
= 30° or 210°, obviously this one is in the third quadrant

Thus z = (2√21)∠210°

Answer 2

Convert to polar form.

-3√7 - i√21

First find r.

r² = (-3√7)² + (-√21)² = 63 + 21 = 84
r = √84 = 2√21

Now find θ. Since both terms are negative θ lies in the third quadrant.

tanθ = -√21 / (-3√7) = √21/√63 = 1/√3

θ = 7π/6

(r, θ) = (2√21, 7π/6)

The polar form of the complex number is therefore:

z = r cos(θ) + ir sin(θ)
z = (2√21)cos(7π/6) + i(2√21)sin(7π/6)
z = (2√21)(-√3/2) + i (2√21)(-1/2)
z = -√63 - i√21
z = -3√7 - i√21