Question

URGENT!!!!!!! Find all seventh roots of unity and sketch them on the axes below.

Answer 1

The 7th roots are :
`$ 1, (2 \pi)/(7), (4 \pi)/(7), (6 \pi)/(7), (8 \pi)/(7), (10 \pi)/(7), (12 \pi)/(7)$`

Explanation:

The roots of unity are evenly spread around the unit circle.

The roots of unity can be find by using the relation

`$ 1= 1 ( \cos 0 ^\circ +i \sin ^\circ )$`

`$\sqrt[n]{1} = 1 [\cos ((2k \pi)/(n)})+ i \sin ((2k \pi)/(n)) ] $`

Now z be a polynomial.

`$z^7=1 \Rightarrow z = 1^{(1)/(7)}$`

therefore, cos 0 = 1.

`$ z = \cos (2k \pi)^{(1)/(7)}$`
`$ (\cos \theta)^n = \cos n \theta $`

`$ z = \cos (2k \pi)/(7) $`

Now, for k=0, z = 1

`$ k=1 \Rightarrow z = \cos (2 \pi)/(7) = \cos 3 (2 \pi)/(7)+ i \sin (2 \pi)/(7)$`

`$ k=2 \Rightarrow z = \cos (4 \pi)/(7) = \cos (4 \pi)/(7)+ i \sin (4 \pi)/(7)$`

`$ k=3 \Rightarrow z = \cos (6 \pi)/(7) = \cos (6 \pi)/(7)+ i \sin (6 \pi)/(7)$`

`$ k=4 \Rightarrow z = \cos (8 \pi)/(7) = \cos (8 \pi)/(7)+ i \sin (8 \pi)/(7)$`

`$ k=5 \Rightarrow z = \cos (10 \pi)/(7) = \cos (10 \pi)/(7)+ i \sin (10 \pi)/(7)$`

`$ k=6 \Rightarrow z = \cos (12 \pi)/(7) = \cos (12 \pi)/(7)+ i \sin (12 \pi)/(7)$`

`$ k=7 \Rightarrow z = \cos (14 \pi)/(7) = \cos (14 \pi)/(7)+ i \sin (14 \pi)/(7)$`

Then the 7th roots are :
`$ 1, (2 \pi)/(7), (4 \pi)/(7), (6 \pi)/(7), (8 \pi)/(7), (10 \pi)/(7), (12 \pi)/(7)$`