De Moivre’s Theorem

One of the most helpful and fundamental theorems in the study of complex numbers is called DeMoivre’s Theorem.

 This theorem establishes a connection between complex numbers and trigonometry. 

It is also helpful for discovering correlations between the functions of numerous angles that are calculated using trigonometry. 

The DeMoivre Theorem is also known as “De Moivre’s Identity” and “De Moivre’s Formula.”

 Both of these names refer to the same thing. 

The famous mathematician De Moivre, who made significant contributions to the field of mathematics, primarily in the areas of theory of probability and algebra, was honoured with the naming of this theorem after his name.

De Moivre’s Formula

The following is true for any real integer x: [Mathematical Statement] 

(cos x + i sin x)n = cos(nx) + isin (nx)

OR

Where ” n ” is a positive integer, ” i” is the imaginary portion, and i = √-1 

Also assume i2 = -1.

Remark: The result may be proved to be true even if n is even when n is a negative integer.

n is what’s known as a rational number.

Consider a complex number written in polar form as z = r (cos + i sin ). 

This will help you grasp how De Moivre’s formula works.

 Let’s multiply this by powers 2 and see what the results are.

z2=[r(cosθ+isinθ)]2

=r2(cosθ+isinθ)(cosθ+isinθ) 

=r2(cosθ cosθ+i sinθ cosθ+i sinθ cosθ- sin θ sinθ) 

=r2[(cosθ cosθ−sinθ sinθ)+i(sinθ cosθ+sinθ cosθ)]

=r2(cos2θ+i sin 2θ)

By extending it manually, we can see that (r (cos + i sin ))3 = r3 (cos 3θ  + i sin 3θ). 

This is similar to the previous example. 

The De Moivre formula is constructed around this fundamental premise.

 If a complex number, written in polar form as r (cosθ + i sinθ ), is multiplied by some power ‘n’ (where n is an integer), the modulus of the result is rn, and the argument of the result is n.

 If r is a complex number written in polar form, then rn is the modulus of the result.

Therefore, the De Moivre Formula is as follows:

(r (cos nθ + i sin nθ)n = rn (cos nθ + i sin nθ), where n €Z.

The theorem of de Moivre

It says that for any number n, the expression

 (cos θ+ i sin θ )n = cos (nθ) + i sin (nθ) holds true. 

Using Euler’s formula, which is presented in the following sentence, we can simply demonstrate this point.

De Moivre’s Theorem Proof

In order to demonstrate De Moivre’s Theorem, you should use mathematical induction.x

We know, (cos x + i sin x)n = cos(nx) + i sin(nx) …

First, assuming that n is equal to one, we obtain (cos x + i sin x).

1 = cos(1x) + i sin(1x) = cos(x) + I sin (x)

Which is a valid point.

The second step is to make the assumption that the formula for n = k is correct.

(cos x + i sin x)

k = cos(kx) + i sin(kx) …. 

Step 3: Convince yourself that the conclusion is valid for the equation n = k + 1.

(cos x + i sin x)k+1 = (cos x + isin x)k (cos x + i sin x) 

= (cos (kx) + i sin (kx)) (cos x + i sin x) 

= cos (kx) cos x − sin(kx) sinx + i (sin(kx) cosx + i cos(kx) sinx) 

= cos {(k+1)x} + i sin {(k+1)x} 

=> (cos x + i sin x) k+1 = cos {(k+1)x} + i sin {(k+1)x}

The conclusion can therefore be validated.

Since the theory holds true when n is equal to one and when n is equal to k plus one, it holds true when n is less than one.

DeMoivre corollary 

The DeMoivre Formula is really quite remarkable.

 It states that if you take a number that is on the unit circle (that is, with length 1) and multiply it by itself, that number will simply rotate around the unit circle by the angle t. 

This is because the first argument is the angle. 

The vector will rotate by an additional t degrees whenever you perform the operation of multiplying the number by itself.

 To put it another way, the output of the power operator in this scenario is a straightforward rotation.

Conclusion

Despite the fact that Abraham de Moivre never mentioned the formula in any of his writings, it bears his name.

 cis x is an abbreviation that is sometimes used in place of the expression cos x + i sin x.

The formula is significant because it establishes a connection between trigonometry and complex numbers.

 It is possible to derive useful expressions for cos nx and sin nx in terms of cos x and sin x if one first expands the left-hand side and then compares the real and imaginary parts under the assumption that x is real. 

These expressions can be found by expanding the left-hand side.

The formula, in its current form, cannot be applied to powers of n that are not integers.

Having said that, there are generalisations of this formula that can be applied to other exponents and be valid.

 These can be put to use to give explicit expressions for the nth roots of unity, which are complex numbers z such that

 zn = 1 in mathematical parlance.