Question

Find all the third roots of i and plot them on a Argand diagram. (please also state which theorems was used) .

Answer 1

Given complex number -

z=i

$Real(z)=x=0\\ Img(z)=y=1$

$\phi=arg(z)=tan^{-1}(\dfrac{1}{0})=tan^{-1}(\infty)=\dfrac{\pi}{2}$

The above complex number in polar form is-

z$=cos\dfrac{\pi}{2}+isin\dfrac{\pi}{2}$

To find its cube root we take-

$z_k=\sqrt[3]{z}=\sqrt[3]{|z|}(cos {\frac {\phi+2\pi k} 3}+isin{\frac {\phi+2\pi k} 3}), k=0,1,2.$

So, let's find all $z_k$

$z_0=(cos {\frac {\pi} 6}+isin{\frac {\pi} 6})$

$z_1=(cos {\frac {5\pi} 6}+isin{\frac {5\pi} 6})$

$z_2=(cos {\frac {3\pi} 2}+isin{\frac {3\pi} 2})$

Here Demovier's theorem is used.

Argand diagram is-