Can someone help me with this algebra question (imaginary numbers)?

Question

The answer to this is a), all i dont understand is how to solve for the Theta (θ) ? can someone please explain that to me, Thank You!

Answer 1

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ),

where, r = | z | = √(a² + b²).

if a > 0, then θ = tan^{- 1}(b / a), and

if a < 0, then θ = tan^{- 1}(b / a) + π or θ = tan^{- 1}(b / a) + 180^o.

 

The complex number is z = 4(1 + i√3) = 4 + 4√3i.

The polar form of a complex number z = a + bi is z = r (cos θ + i sin θ).

Here, a = 4 > 0 and b = 4√3.

So, first find the absolute value of r .

r = | z | = √(a ² + b ²)

            = √[ 4² + (4√3)² ]

            = √[16 + 48]

            = √64

            = 8.

Now find the argument θ.

Since a = 4 > 0, use the formula θ = tan^{- 1}(b / a).

θ = tan^{- 1}[ (4√3)/4 ]

θ = tan^{- 1}(√3)

θ = 60⁰

θ = 60 * π/180 radians

θ = π/3 radians.

The polar form of z is about  8[ cos (π/3) + i sin (π/3) ].

By euler's equation, we obtain the polar form as : z = re ^iθ.

⇒ z = 8e^i(π/3) .

w = z^1/3 = [ 8e^i(π/3) ]^1/3 = 2e^(π/9)i .

Therefore, w = 2e^(π/9)i .

Option d is the correct choice.