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Definition, Formula, Solved Example Problems - de Moivre's Theorem | 12th Mathematics : UNIT 2 : Complex Numbers

Chapter: 12th Mathematics : UNIT 2 : Complex Numbers

de Moivre's Theorem

Now let us apply de Moivre’s theorem to simplify complex numbers and to find solution of equations.

de Moivre's Theorem

de Moivre’s Theorem 

Given any complex number cosθ sinθ and any integer n,

(cosθ sinθ )n cos nθ sin nθ .


Corollary

(1) (cosθ sinθ )n  = cos nθ sin  

(2) (cosθ sinθ )-n  = cos nθ sin  

(3) (cosθ sinθ )-n = cos nθ sin  

(4) sinθ cosθ (cosθ 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  .

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 = (cosθ + 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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12th Mathematics : UNIT 2 : Complex Numbers : de Moivre's Theorem | Definition, Formula, Solved Example Problems

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12th Mathematics : UNIT 2 : Complex Numbers


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