Complex numbers question help... (1 Viewer)

[jack2808]

hey,I'm not confident of the process of solving questions Z^n=k (de moivres theorem), i understand de movires theorem in roots of unity questions but dont understand a question such as Z^3+ 8i = 0
any help would be much appreciated

 

[Spiritual Being]

Z^3 + 8i =
z^3 -8i^3 =
(z-2i)(z^2 +2iz + 4i^2) =
(z -2i)(z^2 +2iz -4)





z^3+8i=0

z^3=-8i

r=|-8i|=sqrt(0^2+(-8)^2)

=sqrt(64)=8

z=r^(1/n)[cos((t+360k)/n)+ i sin((t+360k)/n)], k=0,1,2,..,n-1

cost=x/r=0/8=0

sint=y/r=-8/8=-1

Then t=-90

z=8^(1/3)[cos((-90+360k)/3)+ i sin((-90+360k)/3)], k=0,1,2

z1=2[cos(-90/3)+ i sin(-90/3)]

=2[cos (-30 )+ i sin(-30)] , for k=0

Do the same by substituting k=1 and k=2 in the formula to get the other two roots.


maths = love.

 

[deswa1]

<img src="http://latex.codecogs.com/gif.latex?z^3=-8i\\ z^3=8cis\frac{-\pi+4k\pi}{2}\\ z=2cis\frac{-\pi+4k\pi}{6}" title="z^3=-8i\\ z^3=8cis\frac{-\pi+4k\pi}{2}\\ z=2cis\frac{-\pi+4k\pi}{6}" />

 

[jack2808]

thanks heaps guys !! maths ≥ love