What is the De’ Moivre’s Theorem?

What is the De’ Moivre’s Theorem?

(1) If n is any rational number, then (cos θ + i sin θ)n = cos nθ + i sin nθ.
What is the De' Moivre's Theorem 1(2) If z = (cos θ1 + i sin θ1) (cos θ2 + i sin θ2) (cos θ3 + i sin θ3)………… (cos θn + i sin θn)
then z = cos (θ1 + θ2 + θ3 + ……… + θn) + i sin (θ1 + θ2 + θ3 + ……… + θn)
where θ1 + θ2 + θ3 + ……… + θn R.
(3) If z = r(cos θ + i sin θ) and n is a positive integer, then
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Deductions:

If n Q, then
(i) (cos θ − i sin θ)n = cos nθ − i sin nθ
(ii) (cos θ + i sin θ)-n = cos nθ − i sin nθ
(iii)  (cos θ − i sin θ)-n = cos nθ + i sin nθ
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This theorem is not valid when n is not a rational number or the complex number is not in the form of cos θ + i sin θ.

Roots of a complex number

(1) nth roots of complex number (z1/n)
Let z = r(cos θ + i sin θ) be a complex number. By using De’moivre’s theorem nth roots having n distinct values of such a complex number are given by
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Properties of the roots of z1/n
(i) All roots of z1/n are in geometrical progression with common ratio e2πi/n.
(ii) Sum of all roots of z1/n is always equal to zero.
(iii) Product of all roots of z1/n = (−1)n-1 z.
(iv) Modulus of all roots of z1/n are equal and each equal to r1/n or |z|1/n
(v) Amplitude of all the roots of z1/n are in A.P. with common difference 2π/n.
(vi) All roots of z1/n lies on the circumference of a circle whose centre is origin and radius equal to |z|1/n. Also these roots divides the circle into n equal parts and forms a polygon of n sides.

(2) The nth roots of unity
The nth roots of unity are given by the solution set of the equation
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Properties of nth roots of unity

(i) Let What is the De' Moivre's Theorem 6 the nth roots of unity can be expressed in the form of a series i.e., 1, α, α2, ….. αn-1. Clearly the series is G.P. with common ratio α i.e., ei(2π/n).
(ii) The sum of all n roots of unity is zero i.e., 1, α, α2, ….. αn-1 = 0.
(iii) Product of all n roots of unity is (−1)n-1.
(iv) Sum of pth power of n roots of unity
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(v) The n, nth roots of unity if represented on a complex plane locate their positions at the vertices of a regular polygon of n sides inscribed in a unit circle having centre at origin, one vertex on positive real axis.
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If one of the complex root is then other root will be or vice-versa.

Properties of cube roots of unity
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(iv) The cube roots of unity, when represented on complex plane, lie on vertices of an equilateral triangle inscribed in a unit circle having centre at origin, one vertex being on positive real axis.
(v) A complex number a + ib, for which |a : b|= 1 : √3 or √3 : 1, can always be expressed in terms of i, ω, ω2.
(vi) Cube root of –1 are −1, −ω, −ω2.

(4)  Fourth roots of unity : The four, fourth roots of unity are given by the solution set of the equation x4 − 1 = 0.
⇒ (x2 – 1)(x2 + 1) = 0 ⇒ x = ±1, ±i
Fourth roots of unity are vertices of a square which lies on coordinate axes.