Write the complex number z=(-1-√3i)^19 in polar form: z=r(cosθ+isinθ) where r =? and θ=? The angle should satisfy 0≤θ<2π

Question

As the title states. I went through this problem myself and got r = 524288 and θ = 2π/3 , but that was not correct.

Answer 1

Ignoring the exponent, you can first find z₀ where r = sqrt(a² + b²) and tan^-1(y / x) = θ. Remember, θ should reflect the quadrant we're in, the 3rd.

z₀​ = 2(cos(4π/3)​ + isin(4π/3)​)

Then, apply De Moivre's Theorem (the formula should be in your notes/textbook):

n = 19

19 * 4π/3 = 76π/3

24π + 4π/3 = 76π/3 (so 4π/3 is coterminal with 76π/3 and is within 0≤θ<2π)

z = 2¹⁹(cos(4π/3)​ + isin(4π/3)​)

r = 524288

θ = 4π/3

Hope this was helpful.

Answer 2

Isn't z=[2(cos(4π/3)+isin(4π/3))]¹⁹

so the angle of z is: 19(4π/3)=76π/3 which is the same as 25π+π/3, same as π+π/3=4π/3