What is DeMoivre’s Theorem?

DeMoivre’s theorem is the most important and useful theorem which connects trigonometry and complex numbers.

DeMoivres theorem is also known as the DeMoivre identity and DeMoivre formula. The name of the theorem is after the name of the great mathematician DeMoivre.

Statement:- (cosθ+isinθ)ⁿ=cosnθ+isinnθ

∀ n∈Z

Proof:-

When n=0

(Cosθ+isinθ)⁰=cos0θ+isin0θ

(Cosθ+isinθ)⁰=cos0+isin0 –>(1)

(Cosθ+isinθ)⁰ =1

Cos0=1

Sin0=0

Put in equation (1)

1=1+0

1=1

The theorem is true for n=0–>(a)

When n is a positive integer

We prove the theorem by mathematical induction method.

Let n=1

(Cosθ+isinθ)ⁿ=cosnθ+isinnθ

(Cosθ+isinθ)¹=cos1θ+isin1θ

(Cosθ+isinθ)¹=cosθ+isinθ

It is true for n=1

Theorem is true for n=1

Suppose theorem is true for n=k

(Cosθ+isinθ)ⁿ=cosnθ+isinnθ

(Cosθ+isinθ)^k=coskθ+isinkθ

Multiply by Cosθ+isinθ

(Cosθ+isinθ)^k(Cosθ+isinθ )¹=(coskθ+isinkθ)( Cosθ+isinθ)

(Cosθ+isinθ)^k+1=

( coskθ Cosθ+  coskθ isinθ) +

(isinkθ Cosθ+(i)²sinkθsinθ )

(Cosθ+isinθ)^k+1=

(coskθ Cosθ+(i)²sinkθsinθ) +

 (Coskθ isinθ+ isinkθ Cosθ)

(Cosθ+isinθ)^k+1=

(coskθ Cosθ-sinkθsinθ)

+i( Coskθ sinθ+ sinkθ Cosθ)

            ∴ (i)²=-1

(Cosθ+isinθ)^k+1=

(coskθ Cosθ-sinkθsinθ) i(sinkθ Cosθ+ Coskθ sinθ)

By using

Cos (α+β)=cosαcosβ-sinαsinβ

Sin(α+β)=sinαcosβ+cosαsinβ

(Cosθ+isinθ)^k+1=

cos(k+1)θ+isin(k+1)θ

Theorem is true for n=K+1

Theorem is a true for all positive integer –>(b)

When n is negative integer

Let n=-m

(Cosθ+isinθ)ⁿ=cosnθ+isinnθ

L.H.S

(Cosθ+isinθ)ⁿ =(Cosθ+isinθ)^-m

(Cosθ+isinθ)ⁿ   = 1/(Cosθ+isinθ)^m

(Cosθ+isinθ)ⁿ=1/(Cosmθ+isinmθ)

(Cosθ+isinθ)ⁿ=1/(Cosmθ+isinmθ) X ((Cosmθ-isinmθ) /(Cosmθ-isinmθ))

(Cosθ+isinθ)ⁿ= (Cosmθ-isinmθ) /(Cos²mθ+sin²mθ)

                                                

∴ Cos²mθ+sin²mθ=1

(Cosθ+isinθ)ⁿ= (Cosmθ-isinmθ)

(Cosθ+isinθ)ⁿ=Cos(-m)θ+isin(-m)θ

(Cosθ+isinθ)ⁿ= cosnθ+isinnθ –>(c)

So Theorem is true for all negative integers

By a,b and c

The theorem is true for all integer n