de Moivre's Theorem
de Moivre’s Theorem
Given any complex number cosθ + i sinθ and any integer n,
(cosθ + i sinθ )n = cos nθ + i sin nθ .
Corollary
(1) (cosθ - i sinθ )n = cos nθ - i sin nθ
(2) (cosθ + i sinθ )-n = cos nθ - i sin nθ
(3) (cosθ - i sinθ )-n = cos nθ + i sin nθ
(4) sinθ + i cosθ = i (cosθ - i sinθ ) .
Now let us apply de Moivre’s theorem to simplify complex numbers and to find solution of equations.
Example 2.28
If z = (cosθ + i sinθ ) , show that zn + 1/ zn = 2 cos nθ and zn – [1/ zn] = 2i sin nθ .
Solution
Let z = (cosθ + i sinθ ) .
By de Moivre’s theorem ,
zn = (cosθ + i sinθ )n = cos nθ + i sin nθ
Example 2.29
Similarly,
Solution
Example 2.30
Solution
Example 2.31
Simplify
(i) (1+ i)18
(ii) (-√3 + 3i)31 .
Solution
(i) (1+ i)18
Let 1+ i = r (cosθ + i sinθ ) . Then, we get
(ii) (-√3 + 3i)31 .
Let -√3 + 3i = r (cosθ + i sinθ ) . Then, we get
Raising power 31 on both sides,
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