- For any real number x, e
^{ix}= cos x + i sin x - Let z be a non zero complex number; we can write z in the polar form as,

z = r(cos θ + i sin θ) = r e^{iθ}, where r is the modulus and θ is argument of z. **Important point**

Multiplying a complex number z with e^{iα}gives

z *e^{iα}= re^{iθ}X e^{iα}= re^{i(α + θ)}

The resulting complex number re^{i(α + θ)}will have the same modulus r and argument (α + θ).

__Case I__It states that for any integer n,

(cos θ + i sin θ)^{n}= cos (nθ) + i sin (nθ)

__Case II__if n is of the form p/q where p, q are integers and q > 0

then

(cos θ + i sin θ)^{n}=cos (2k π+ θ)p/q + sin (2k π+ θ)p/q

Where k=0,1,2,...q-1

This can be easily proved using Euler's formula as shown below.

We know that, (cos θ + i sin θ) = e

(cos θ + i sin θ)

Therefore,

e

(cos θ + i sin θ)

(cos θ + i sin θ)

(cos θ + i sin θ)

(cos θ + i sin θ)

=(cos θ + i sin θ)

=[cos (mθ) + i sin (mθ)](cos θ + i sin θ)

=[cos(mθ) cosθ -sin(mθ)sin θ] +i[cos (mθ)sin θ + sin (mθ)cos θ]

=cos (m+1)θ + i sin (m+1)θ

So, theorem holds good for n=m+1

So, by principle of Mathematical Induction

(cos θ + i sin θ)

For negative integers, we can write like n=-m where m is positive number

(cos θ + i sin θ)

Now m being positive, we already proved above

=1 /cos (mθ) + i sin (mθ)

Multiplying the Conjugate complex on Numerator and Denominator we get,

=cos (mθ) - i sin (mθ)

=cos (-mθ) +i sin (-mθ)

=cos (nθ) + i sin (nθ)

- De Moivre's theorem is used to calculate the root of Complex Numbers
- For a complex number z=a + ib which can be expressed as polar form as = r(cos θ + sin θ)

So z^{1/n}= r^{1/n}(cos θ + sin θ)^{1/n}

Now from De Moivre's theorem, we know that

(cos θ + i sin θ)^{1/n}=cos (2k π+ θ)/n + sin (2k π+ θ)/n

Where k=0,1,2,...n-1

- By putting value of k=0,1,2,...n-1, we can find the root values

- What is complex numbers
- |
- Algebra Of complex Numbers
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- Conjugate of Complex Numbers
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- Modulus of complex numbers
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- Argand Plane
- |
- Polar Representation of the complex number
- |
- Rotation of Complex Number
- |
- Identities for Complex Numbers
- |
- Eulers formula and De moivre's theorem
- |
- Cube Root of unity

Class 11 Maths Class 11 Physics

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