De Moivre'S Theorem And Nth Roots Worksheet Page 5
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6Example 8
Find the cube roots of unity and plot them on an Argand diagram.
Answer:
Since 1 can be written in polar form as
1 = cos 2 + i sin 2
1 = cos 4 + i sin 4
1 = cos 6 + i sin 6
the cube roots of unity are
z =
=
z =
=
z =
= 1
On the Argand diagram
Example 9
Find the fourth roots of unity and plot them on an Argand diagram.
The solutions are
z = cos 0 + i sin 0 ,
, cos + i sin and
i.e. z = 1, -1, i and -i
On an Argand diagram this gives
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