De Moivre Formula

De Moivre Formula

Despite the fact that some students may find Mathematics to be a difficult academic subject, mathematicians from all over the world have made major contributions to it over the course of centuries. The NCERT textbooks are structured in accordance with students’ requirements and expectations and were developed with knowledge of what students’ brains can absorb at various ages. The NCERT textbooks are written in order to foster students’ intellectual development. To support students in their academic endeavours while taking into account the structure of NCERT books, the Extramarks online learning platform created NCERT solutions. The knowledge of Mathematics is used in a number of different academic fields including Engineering, Physics, and Chemistry. Mathematical principles must be understood for learning many of these topics. Mathematics is also taught as an “instrumental subject” to aid students in learning other academic subjects, despite the fact that it is occasionally taught in integrated courses in some other curricula.

Students must know the basics of Algebra to understand the questions of De Moivre Formula.

De Moivre Formula, which unites complex numbers and trigonometry, is one of the most significant and practical theorems in the study of complex numbers. Obtaining correlations between trigonometric functions of various angles is also helpful. “De Moivre’s Identity” and “De Moivre Formula” are other names for DeMoivre’s Theorem. The famous mathematician De Moivre, who made numerous contributions to the subject of Mathematics, particularly in the areas of Probability Theory and Algebra, is honoured in the name of the theorem.

De Moivre Formula is a significant mathematical theme. Complex numbers are related to De Moivre Formula. Similar to how students can increase the power of any binomial, students can improve the power of a complex number. However, De Moivre Formula makes it considerably easier to determine a complex number’s power. The complex number must first be changed into polar form before using De Moivre Formula.

What Is De Moivre’s Formula?

Consider the complex number z = r (cos I + I sin ) to better grasp De Moivre Formula. Students can multiply this by two to see what occurs.

By manually extending it, one can see that (r (cos + I sin))3 = r3 (cos + I sin 3). The De Moivre Formula’s main concept is as follows. When a complex number (in polar form) is raised to the power ‘n’ (where n is an integer), the modulus and argument of the result are each equal to rn. De Moivre Formula is as a result:

(r (cos θ + i sin θ))n = rn (cos nθ + i sin nθ), where n ∈ Z

Complex numbers, which are frequently expressed in rectangular or standard form, are those with the formula a + ib, where a and b are real numbers and I (iota) is the imaginary component and stands for (-1). For instance, 10 + 5i is a complex number where 10 denotes the real element and 5i denotes the imaginary component. The values of a and b determine whether they are entirely real or entirely fake. When a = 0, the result is a, which is a strictly real number. When b = 0, the result is ib, which is a completely imaginary number. 

De Moivre’s Theorem for Fractional Power

Now students must learn to prove De Moivre’s theorem by the principle of mathematical induction.

Here, assume that S(n) : (r (cos θ + i sin θ))n = rn (cos nθ + i sin nθ).

Step 1: To prove 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)

Thus, S(n) is true for n = 1.

Step 2: Assume that S(n) is true for some natural number n = k. Then

(r (cos θ + i sin θ))k = rk (cos kθ + i sin kθ)

Step 3: To prove S(n) for n = k + 1.

LHS = (r (cos θ + i sin θ))k + 1

= (r (cos θ + i sin θ))k • (r (cos θ + i sin θ))

= rk (cos kθ + i sin kθ) •   (r (cos θ + i sin θ)) (By Step 2)

= rk + 1 [(cos kθ cos θ – sin kθ sin θ) + i (cos kθ sin θ + sin kθ cos θ)]

= rk + 1 [ cos (kθ + θ) + i sin (kθ + θ) ]

= rk + 1 [ cos (k + 1)θ + i sin (k + 1)θ ]

= RHS

So S(n) is true for n = k + 1.

As a result, according to the mathematical induction principle, S(n) holds true for all values of n.

Students are aware that the formula for cos + I sin is cis. The De Moivre Formula can be expressed as follows:

(r cis θ)n = rn cis nθ, where n ∈ Z

Solved Examples Using De Moivre’s Formula

Here are a few solved questions for students with the help of the De Moivre Formula. Students must solve each and every question regarding this very formula.

Find the value of (1 – √3 i)5 using the De Moivre Formula.

Let z = 1 – √3 i = a + ib.

Its modulus is, r = √(a2 + b2) = √(1+3) = 2.

α = tan-1 |b/a| = tan-1 √3 = π/3.

Since a > 0 and b < 0, θ is in the 4th quadrant. So 

θ = 2π – π/3 = 5π/3

Thus, z = r (cos θ + i sin θ) = 2 (cos 5π/3 + i sin 5π/3)

Now, z5 = (2 (cos 5π/3 + i sin 5π/3))5

By De Moivre Formula,

z5 = 25 (cos 25π/3 + i sin 25π/3)

= 32 (1/2 + √3/2 i)

= 16 + 16 √3 i

Answer: (1 – √3 i)5 = 16 + 16 √3 i.

Practice Questions on Daily Compound Interest Formula

A complex number must first be converted to its polar form, which includes the modulus and argument as parts, in order to be expanded according to the exponent supplied. The De Moivre Formula is then applied, and it states:

According to De Moivre’s Formula, for all real values of a number, let’s say x

(cos x + isinx)n = cos(nx) + isin(nx)

where any integer n is used