find all complex roots of w=16(cos(4π /3)+ i sin (4π /3)). write your answer in rectangular form.?

[匿名]

2020-05-12 8:04 am

find all complex roots of w=16(cos(4π /3)+ i sin (4π /3)).

write your answer in rectangular form.

okay this is what i got below please correct me if i'm wrong in any of them;

z0= 4(cos 60 degrees+ i sin 60 degrees)

z2= 4(cos 150 degrees+ i sin 150 degrees)

z3= 4(cos 240 degrees+ i sin 240 degrees)

I DO NOT KNOW HOW TO CONVERT TO RECTANGULAR FORM. PLEASE HELP!

回答 (1)

用戶26240

2020-05-12 8:34 am

✔ 最佳答案

w = 16 [cos(4π/3) + i sin(4π/3)]
w = 16 [cos(240°) + i sin(240°)]

By De Moivre's theorem:
w^(1/2) = (16)^(1/2) [cos(240°) + i sin(240°)]^(1/2)
zₖ = 4 [cos{(360k° + 240°)/2} + i sin{(360k° + 240°)/2}]
where k = 0, 1

When k = 0:
zₒ = 4 [cos(120°) + i sin(120°)]
zₒ = 4 [cos(180° - 60°) + i sin(180° - 60°)]
zₒ = 4 [-cos(60°) + i sin(60°)]
zₒ = 4 [-(1/2) + (√3/2)i]
zₒ = -2 + 2√3i

When k = 1:
z₁ = 4 [cos(300°) + i sin(300°)]
z₁ = 4 [cos(360° - 60°) + i sin(360° - 60°)]
z₁ = 4 [cos(60°) - i sin(60°)]
z₁ = 4 [(1/2) - (√3/2)i]
z₁ = 2 - 2√3i

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