# 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 =

• 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 ,    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 + i sin         k = 0, 1, 2, ……, n – 1

Let α = cos + i sin

When k = 2

=             [By De–Moivre’s Theorem]

= α2

when k = 3

= α3

Similarly, when k = t

Then, = αt

∴ The roots are 1, α, α2, ……, αn –1

Sum of the Roots

1 + α + α2 + .... + αn – 1 = = 0         ( αn = 1)

and

Thus the sum of the roots of unity is zero.

Product of the Roots

1.α.α2. .......... αn – 1 = = = cos{π(n – 1)} + i sin{π(n – 1)}

If n is even =–1

If n is odd =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.