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 cos^n–2θ sin²θ + nc⁴ cos^n–4θ sin⁴θ + ………

sin ⁿθ = ⁿc₁ cos^n–1θ sinθ – ⁿc₃ cos^n–3θ sin³θ + ⁿc₅ cos_n–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

z^1/n = r^1/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.

n^th 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, = α^t

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

Sum of the Roots

1 + α + α² + .... + α^{n – 1} = = 0         ( αⁿ = 1)

and

Thus the sum of the roots of unity is zero.

Product of the Roots

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

If n is even =–1

If n is odd =1