De Moivre'S Theorem And Nth Roots Worksheet Page 3
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5
6In this way the nth roots of any complex number can be found.
Example 5
Find the cube roots of z = 64(cos 30° + i sin 30°)
Answer:
This is in polar form. Use 2 = 360° and 4 = 720°
z = 64(cos 30° + i sin 30°)
z can also be written as
z = 64{cos (30 + 360)° + i sin (30 + 360)°}
and
z = 64{cos (30 + 720)° + i sin (30 + 720)°}
Since
, the cube roots of z are
4(cos 10° + i sin 10°), 4(cos 130° + i sin 130°), 4(cos 250° + i sin 250°)
Example 6
Find the fourth roots of 81i, that is of
z =
z =
z =
z =
= 3
The fourth roots of
are
3
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