The 7th roots of 1 are the numbers [tex]z[/tex] such that [tex]z^7=1[/tex]. They take the form
[tex]z=1^{1/7}\left(\cos\dfrac{(0+360n)^\circ}7+i\sin\dfrac{(0+360n)^\circ}7\right)[/tex]
with [tex]n[/tex] ranging from 0 to 6. This is because, when you raise any of these [tex]z[/tex] to the 7th power, by DeMoivre's theorem we get
[tex]z^7=\left(1^{1/7}\right)^7\left(\cos(0+360n)^\circ+i\sin(0+360n)^\circ\right)\implies z^7=1[/tex]
Then the roots themselves are
[tex]z_1=\cos0^\circ+i\sin0^\circ[/tex]
[tex]z_2=\cos\left(\dfrac{360}7\right)^\circ+i\sin\left(\dfrac{360}7\right)^\circ[/tex]
[tex]z_3=\cos\left(\dfrac{720}7\right)^\circ+i\sin\left(\dfrac{720}7\right)^\circ[/tex]
[tex]z_4=\cos\left(\dfrac{1080}7\right)^\circ+i\sin\left(\dfrac{1080}7\right)^\circ[/tex]
[tex]z_5=\cos\left(\dfrac{1440}7\right)^\circ+i\sin\left(\dfrac{1440}7\right)^\circ[/tex]
[tex]z_6=\cos\left(\dfrac{1800}7\right)^\circ+i\sin\left(\dfrac{1800}7\right)^\circ[/tex]
[tex]z_7=\cos\left(\dfrac{2160}7\right)^\circ+i\sin\left(\dfrac{2160}7\right)^\circ[/tex]
