De moivre's theorem

De moivre's theorem definition & Example 

If n is any integer, then (cos θ + i sin θ)n = cos nθ + i sin nθ. This is known as De Movre’s Theorem. Check out Maths Formulas and NCERT Solutions for class 12 Maths prepared by Physics Wallah. 

Remarks:

  • Writing the binomial expansion of (cos θ + i sin θ)n and equating the real part to cos nθ and the imaginary part to sin nθ, we get

cos nθ = cosn θ – nc2 cosn–2θ sin2θ + nc4 cosn–4θ sin4θ + ………

sin nθ = nc1 cosn–1θ sinθ – nc3 cosn–3θ sin3θ + nc5 cosn–5θ sin5θ + ………

⇒    tan nθ = De moivre's theorem

  • If n is rational number, then one of the values of (cos θ + i sin θ)n is cos nθ + i sin nθ. Let n = p/q, where p and q are integers (q > 0)  and p, q have no common factor. Then (cos θ + i sin θ)n has q distinct values, one of which is cos nθ + i sin nθ.

  • If z = r (cos θ + i sin θ), and n is a positive integer, then

z1/n = r1/n De moivre's theorem,    k = 0, 1, 2, ……, n –1.

Here if can be noted that any ‘n’ consecutive values of k will serve the purpose.  

APPLICATIONS OF DE MOIVER’S THEOREM

This is a fundamental theorem and has various applications. Here we will discuss few of these which are important from the examination point of view.

nth Roots of Unity 

One very important application of De–Moivre’s Theorem is in solving equation of nth powers in complex number. Let x be the nth root of unity. Then

xn = 1 = cos 2kπ + i sin 2kπ        (where k is an integer)

⇒ x = cos De moivre's theorem + i sin De moivre's theorem        k = 0, 1, 2, ……, n – 1

Let α = cos De moivre's theorem + i sin De moivre's theorem

When k = 2 

De moivre's theorem

= De moivre's theorem            [By De–Moivre’s Theorem]

= α2

when k = 3

De moivre's theorem = α3

Similarly, when k = t

Then, De moivre's theorem = αt

∴ The roots are 1, α, α2, ……, αn –1   


Sum of the Roots

1 + α + α2 + .... + αn – 1 =De moivre's theorem = 0         (De moivre's theorem αn = 1)

De moivre's theorem and De moivre's theorem

Thus the sum of the roots of unity is zero.

Product of the Roots

1.α.α2. .......... αn – 1 = De moivre's theorem = De moivre's theorem= cos{π(n – 1)} + i sin{π(n – 1)}

If n is even De moivre's theorem=–1

If n is odd De moivre's theorem=1

The points represented by n nth roots of unity are located at the vertices of a regular polygon of n sides inscribed in a unit circle having centre at the origin, one vertex being on the positive real axis. Geometrically represented as follows.

De moivre's theorem

 

Talk to Our counsellor