Question

1. There are 7 seventh roots of 1. Find three of the non-obvious ones

Find the angles A in radians. Do not write A = 0, 360 degrees or any coterminal angle to o
those are obvious )
Two roots are z = cos A + i sin A
where A= or A= or A=

Can someone help me for this question I kinda confused pls explain to me step by step?

Answer 1

The 7th roots of 1 are the numbers
`z` such that
`z^7=1`. They take the form

`z=1^(1/7)\left(\cos\frac{(0+360n)^\circ}7+i\sin\frac{(0+360n)^\circ}7\right)`

with
`n` ranging from 0 to 6. This is because, when you raise any of these
`z` to the 7th power, by DeMoivre's theorem we get

`z^7=\left(1^(1/7)\right)^7\left(\cos(0+360n)^\circ+i\sin(0+360n)^\circ\right)\implies z^7=1`

Then the roots themselves are

`z_1=\cos0^\circ+i\sin0^\circ`

`z_2=\cos\left(\frac{360}7\right)^\circ+i\sin\left(\frac{360}7\right)^\circ`

`z_3=\cos\left(\frac{720}7\right)^\circ+i\sin\left(\frac{720}7\right)^\circ`

`z_4=\cos\left(\frac{1080}7\right)^\circ+i\sin\left(\frac{1080}7\right)^\circ`

`z_5=\cos\left(\frac{1440}7\right)^\circ+i\sin\left(\frac{1440}7\right)^\circ`

`z_6=\cos\left(\frac{1800}7\right)^\circ+i\sin\left(\frac{1800}7\right)^\circ`

`z_7=\cos\left(\frac{2160}7\right)^\circ+i\sin\left(\frac{2160}7\right)^\circ`