Hello aspirant

Demoviers theorem is an important part of complex numbers . It is a simple topic and fetches a good amount of questions in the mathematic portion of the Jee main exam.

The theorem can be described as

Case 1

Statement-

If n is any integer, then

(i) (cos θ + i sin θ)n = cos nθ + i sin nθ

(ii) (cos θ1 + i sin θ1) = (cos θ2 + i sin θ2) ......... (cos θn + i sin θn)

= cos θ1 + θ2 + θ3 .................. + θn) + i sin (θ1 + θ2 + ..............  θ3)

Case II

Statement:

If p, q ε and q ≠ 0, then (cos θ + i sin θ)p/q = cos((2kπ + pq)/q) + isin((2kπ+pq/q)  where k = 0,1,2,3,.....,q-1

Note:

Continued product of the roots of a complex quantity should be determined using theory of equations.

Derivation: The Demoivre’s formula can be derived from the Euler’s formula:

eix = cos x +i sin x

The exponential law for integral powers gives Then by Euler’s Formula e

ei(nx) = cos (nx) + i sin (nx)

A more elementary motivation of the theorem comes from calculating

(cos + isin x)2 = cos2x + 2i sin x cosx –sin 2x

= (cos2x – sin2x ) + i(2sinx cosx ) = cos (2x) +i sin(2x)

where the last equality follows from the trigonometric identities.

Hence, this proves the result for n=2