Answer:
For K=0
[tex]cos(\frac{0+2\pi 0}{5})+isin(\frac{0+2\pi 0}{5})\\cos(0)+isin(0)=1+i0[/tex]
For K=1:
[tex]cos(\frac{0+2\pi 1}{5})+isin(\frac{0+2\pi 1}{5})\\cos(2\pi/5)+isin(2\pi/5)=0.3090+i0.9510[/tex]
For K=2:
[tex]cos(\frac{0+2\pi 2}{5} )+isin(\frac{0+2\pi 2}{5} )\\cos(4\pi/5)+isin(4\pi/5)=-0.809+i0.587[/tex]
For K=3:
[tex]cos(\frac{0+2\pi 3}{5} )+isin(\frac{0+2\pi 3}{5} )\\cos(6\pi/5)+isin(6\pi/5)=-0.809-i0.587[/tex]
For K=4:
[tex]cos(\frac{0+2\pi 4}{5} )+isin(\frac{0+2\pi 4}{5} )\\cos(8\pi/5)+isin(8\pi/5)=0.3091-i0.9510[/tex]
Step-by-step explanation:
Fifth Root is given by:
[tex]\sqrt[5]{z}=1+0i[/tex]
The above equation will become:
[tex]z=(1+0i)^5[/tex]
It can be written as:
[tex]z=[cos(0)+isin(0)]^5[/tex]
|z|=1,
According to De-moivre's Theorem:
[tex]z=cos(\frac{0}{5})+isin(\frac{0}{5})\\ z=cos(0)+isin(0)[/tex]
Now, Fifth Roots of unity in standard form a + bi :
[tex]\sqrt[5]{z}=[{cos(0+2\pi k)+isin(0+2\pi k)}]^{1/5}[/tex]
k=0,1,2,3,4
For K=0
[tex]cos(\frac{0+2\pi 0}{5})+isin(\frac{0+2\pi 0}{5})\\cos(0)+isin(0)=1+i0[/tex]
For K=1:
[tex]cos(\frac{0+2\pi 1}{5})+isin(\frac{0+2\pi 1}{5})\\cos(2\pi/5)+isin(2\pi/5)=0.3090+i0.9510[/tex]
For K=2:
[tex]cos(\frac{0+2\pi 2}{5} )+isin(\frac{0+2\pi 2}{5} )\\cos(4\pi/5)+isin(4\pi/5)=-0.809+i0.587[/tex]
For K=3:
[tex]cos(\frac{0+2\pi 3}{5} )+isin(\frac{0+2\pi 3}{5} )\\cos(6\pi/5)+isin(6\pi/5)=-0.809-i0.587[/tex]
For K=4:
[tex]cos(\frac{0+2\pi 4}{5} )+isin(\frac{0+2\pi 4}{5} )\\cos(8\pi/5)+isin(8\pi/5)=0.3091-i0.9510[/tex]
