Given complex number -
z=i
Real(z)=x=0Img(z)=y=1
ϕ=arg(z)=tan−1(01)=tan−1(∞)=2π
The above complex number in polar form is-
z=cos2π+isin2π
To find its cube root we take-
zk=3z=3∣z∣(cos3ϕ+2πk+isin3ϕ+2πk),k=0,1,2.
So, let's find all zk
z0=(cos6π+isin6π)
z1=(cos65π+isin65π)
z2=(cos23π+isin23π)
Here Demovier's theorem is used.
Argand diagram is-
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