De Moivre's Formula: Definition, Derivation, Uses

De Moivre's formula is related to complex numbers. De Moivre's formula is used to expand the power of a complex number. It is the same as the way that we used for expanding the power of a binomial. But we must say that De Moivre's formula is way simpler than the other ones because it simplifies the whole process of finding the power of a complex number. We need to convert the complex number into a polar form for applying De Moivre's formula. Let’s discuss more about De Moivre’s formula, its derivation, uses along with some important questions.

Keywords: De Movire's Formula, History of De Movire's Formula, complex numbers, Derivation of De Movire's formula, theorem for fractional power

Also Read: Euler's Identity

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What is De Movire's Formula?

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De Moivre's formula is related to complex numbers as it is used to expand the power of these numbers. De Movire's formula is also called De Moivre's theorem. As per Mathematics, De Moivre's formula states that each real number x & integer n holds that {\displaystyle {\big}^{n}=\cos nx+i\sin nx, }

i= imaginary unit satisfying the equation as i2=-1

Every complex number can be addressed as a + bi,

A & b are real numbers.

De Moivre's formula proves that the power of a complex number in a polar form is similar to raising the modulus to the same power & then multiplying the argument with the same power accordingly.

For example:

For real numbers;

It is written as, (cos x + I sin x) n = cos(nx) + I sin(nx)

Or

(e^iθ)ⁿ = e^inθ

n= positive integer

I= imaginary part

I= √(-1)

Also, we can assume i2=-1

Theorem for fractional power

( cos θ + I sin θ) 1/n = cos |

The video below explains this:

Complex Numbers Detailed Video Explanation:

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History of De Movire's Formula

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The De Moivre's formula has been created by A French Mathematician, Abraham De Moivre, the formula has been named after him. In 1707 Abraham De Moivre in His A.D. paper in Philosophical transactions of the royal society of London, he deducted a formula, with the help of this formula the recognizable form of De Moivre's formula was obtained.

In the year 1749, Euler, the formula for all the real value of n, by using the Euler’s identity.

Also check:

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What are Complex Numbers?

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A complex number is a number that is expressed in the form of a+bi. Including real & imaginary numbers in themselves. For a number to be a complex number it must have a real & an imaginary number, the real number should come first & the imaginary number should be last.

The standard format for a complex number is a+bi.

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Derivation of De Moivre's Formula

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By Mathematical induction, Here we are using the principle of Mathematical induction for proving the De Moivre's formula;

First, we need to assume that

The mathematical induction,

S(n) : (r (cos θ + I sinθ))ⁿ = rⁿ (cos nθ + i sin nθ).

• Let’s prove that S(n) for n= 1

LHS= (r (cos θ + i sin θ))¹ = r (cos θ + i sin θ)

RHS= r¹ (cos (1)θ + i sin (1)θ)= r (cos θ + i sin θ)

Hence, S(n) is true for n= 1

• Now let’s assume that S(n) is true but for some real numbers n= k.

So, (r(cos θ + i sinθ))^k= r^k (cos kθ + i sin kθ)

• Let’s probe that S(n) for n= k + 1

LHS = (r (cos θ + i sin θ))^k+1 = (r (cos θ + i sinθ))^k •(r (cos θ + i sinθ))

= r^k (cos kθ + I sin kθ) • (r (cos θ + I sin θ)) (same as step 2)

= r^k+1 [( cos kθ + θ) + I sin (kθ + θ) + i(cos kθ sin θ + sin kθ cos θ)]

= r^k+1 [cos (kθ + θ) + I sin (k + 1) θ]

= r ^k+1 [cos (k + 1) θ + I sin (k + 1)θ ]

= RHS

Hence, S(n) is true for n = k+1

Therefore, it is proved by the principle of the mathematical induction that S(n) is true for all the values of n.

As we already know that cos θ + I sin θ can also be written as cis θ.

Hence, the formula can also be written as;

(r cis θ)ⁿ = rⁿ cis nθ,

Whereas, n £ z

From Euler's Identity

The formula comes out as a natural outcome from Euler's identity.

The identity is;

eix = cos x + isin x

e is a natural logarithmic base &

x € R.

As we know from the exponential law that

(ea)b = e ab

Let n € Z

As we have, (e ^ix)ⁿ =e^inx

Hence, nx € R

(cos x + isin x)ⁿ = cos nx + isin nx

Also check: Sin 180 Degrees

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Uses of De Moivre's Formula

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• De Moivre's formula is used for finding out the roots of the complex numbers.
• De Moivre's formula is used to obtain a better relationship between the power of the trigonometric functions & the trigonometric angles.
• It is easy to raise any complex number in both rectangular & polar form to the nth power using De Moivre's theorem.
• If a complex number is given but in a rectangular form then your first need to convert it into a polar form.
• It is easier to solve the equations that are related to complex number roots with the help of De Moivre's theorem.
• De Moivre's theorem is used in obtaining the complex roots of all the polynomial equations.

Also see: Binomial expression

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Things to Remember

• To use De Moivre's formula for complex numbers, make sure that if the number is given in rectangular form then first convert it into a polar form, after that use the formula.
• De Moivre's formula with a radius is [ r(cos θ + I sin θ) ⁿ = rⁿ(cos n θ + I sin nθ), whereas the formula without radius is (cos θ + i sin θ)ⁿ = cos n θ + i sin nθ.
• In De Moivre's formula complex numbers & trigonometry functions are connected.
• A complex number is always made with one real & one imaginary number. The real one must come first & the imaginary one comes at last.

Also check: Real Numbers Formula