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marishachu [46]
3 years ago
15

Express the fifth roots of unity in standard form a + bi. with 1 + 0i

Mathematics
1 answer:
marishachu [46]3 years ago
6 0

Answer:

For K=0

cos(\frac{0+2\pi 0}{5})+isin(\frac{0+2\pi 0}{5})\\cos(0)+isin(0)=1+i0

For K=1:

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

For K=2:

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

For K=3:

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

For K=4:

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

Step-by-step explanation:

Fifth Root is given by:

\sqrt[5]{z}=1+0i

The above equation will become:

z=(1+0i)^5

It can be written as:

z=[cos(0)+isin(0)]^5

|z|=1,

According to De-moivre's Theorem:

z=cos(\frac{0}{5})+isin(\frac{0}{5})\\  z=cos(0)+isin(0)

Now, Fifth Roots of unity in standard form a + bi :

\sqrt[5]{z}=[{cos(0+2\pi k)+isin(0+2\pi k)}]^{1/5}

k=0,1,2,3,4

For K=0

cos(\frac{0+2\pi 0}{5})+isin(\frac{0+2\pi 0}{5})\\cos(0)+isin(0)=1+i0

For K=1:

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

For K=2:

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

For K=3:

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

For K=4:

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

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