Question

Find the six complex sixth roots of unity (1) and plot them.

Short answer

All the six roots are

$$\begin{gathered} z_0=1+0i \\ {z}_1=0.5+i0.866 \\ {z}_2=-0.5+i0.866 \\ {z}_3=-1+0i \\ {z}_4=-0.5-i0.866 \\ {z}_5=0.5-i0.866 \end{gathered}$$

Step-by-step solution

Step 1. Converting into polar form

Firstly we convert the unity into polar form So,

1 = 1 (cos0$°$ +isin0$°$)

Now we express the complex number in its polar form so we can apply the theorem to get the complex roots of the complex number.

$$\begin{gathered} z_n=\sqrt[m]{r}\left[\left(\cos \left(\frac{{\theta }_o}{m}+\frac{2n\mathrm{\pi }}{m}\right)+i\sin \left(\frac{{\theta }_o}{m}+\frac{2n\mathrm{\pi }}{m}\right)\right)\right] \\ Wehave \\ {z}_n=\sqrt[6]{1}\left[\left(\cos \left(\frac{0°}{6}+\frac{360°n}{6}\right)+i\sin \left(\frac{0°}{6}+\frac{360°n}{6}\right)\right)\right] \\ {z}_n=\cos \left(60°n\right)+i\sin \left(60°n\right) \\ (n=0,1,2,3,4,5) \end{gathered}$$

Step.2 Roots of the complex number 

Now we have to find the roots of the complex number by putting the values of n in $z_n$.

role="math" localid="1646680481743" $$\begin{gathered} z_0=\cos (0°)+i\sin (0°) \\ {z}_0=1+0i \\ {z}_1=\cos (60°)+i\sin (60°) \\ {z}_1=0.5+i0.866 \\ {z}_2=\cos (120°)+i\sin (120°) \\ {z}_2=-0.5+i0.866 \\ {z}_3=\cos (180°)+i\sin (180°) \\ {z}_3=-1+0i \\ {z}_4=\cos (240°)+i\sin (240°) \\ {z}_4=-0.5-i0.866 \\ {z}_5=\cos (300°)+i\sin (300°) \\ {z}_5=0.5-i0.866 \end{gathered}$$

Hence by putting the values of n we get all the six complex roots of the complex number.

Step 3. Graph

Now we plotted all the six roots in the graph.