Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 2 De Moivre’s Theorem to solve questions creatively.

## Intermediate 2nd Year Maths 2A De Moivre’s Theorem Formulas

**Statement:**

→ If ‘n’ is an integer, then (cos θ + i sin θ)^{n} = cos nθ + i sin nθ

If n’ is a rational number, then one of the values of

(cos θ + i sin θ)^{n} is cos nθ + i sin nθ

**n ^{th} roots of unity:**

→ n

^{th}roots of unity are {1, ω, ω

^{2}…….. ω

^{n – 1}}.

Where ω = \(\left[\cos \frac{2 k \pi}{n}+i \sin \frac{2 k \pi}{n}\right]\) k = 0, 1, 2 ……. (n – 1).

If ω is a n

^{th}root of unity, then

- ω
^{n}= 1 - 1 + ω + ω
^{2}+ ………… + ω^{n – 1}= 0

**Cube roots of unity:**

→ 1, ω, ω^{2} are cube roots of unity when

- ω
^{3}= 1 - 1 + ω + ω
^{2}= 0 - ω = \(\frac{-1+i \sqrt{3}}{2}\), ω
^{2}= \(\frac{-1-i \sqrt{3}}{2}\) - Fourth roots of unity roots are 1, – 1, i, – i

→ If Z_{0} = r_{0} cis θ_{0} ≠ 0, then the n^{th} roots of Z_{0} are α_{k} = (r_{0})^{1/n} cis\(\left(\frac{2 k \pi+\theta_{0}}{n}\right)\) where k = 0, 1, 2, ……… (n – 1)

→ If n is any integer, (cos θ + i sin θ)^{n} = cos nθ + i sin nθ

→ If n is any fraction, one of the values of (cosθ + i sinθ)^{n} is cos nθ + i sin nθ.

→ (sinθ + i cosθ)^{n} = cos(\(\frac{n \pi}{2}\) – nθ) + i sin(\(\frac{n \pi}{2}\) – nθ)

→ If x = cosθ + i sinθ, then x + \(\frac{1}{x}\) = 2 cosθ, x – \(\frac{1}{x}\) = 2i sinθ

→ x^{n} + \(\frac{1}{x^{n}}\) = 2cos nθ, x^{n} – \(\frac{1}{x^{n}}\) = 2i sin nθ

→ The nth roots of a complex number form a G.P. with common ratio cis\(\frac{2 \pi}{n}\) which is denoted by ω.

→ The points representing n^{th} roots of a complex number in the Argand diagram are concyclic.

→ The points representing n^{th} roots of a complex number in the Argand diagram form a regular polygon of n sides.

→ The points representing the cube roots of a complex number in the Argand diagram form an equilateral triangle.

→ The points representing the fourth roots of complex number in the Argand diagram form a square.

→ The nth roots of unity are 1, w, w^{2},………. , w^{n-1} where w = cis\(\frac{2 \pi}{n}\)

→ The sum of the nth roots of unity is zero (or) the sum of the nth roots of any complex number is zero.

→ The cube roots of unity are 1, ω, ω^{2} where ω = cis\(\frac{2 \pi}{3}\), ω^{2} = cis\(\frac{4 \pi}{3}\) or

ω = \(\frac{-1+i \sqrt{3}}{2}\)

ω^{2} = \(\frac{-1-i \sqrt{3}}{2}\)

1 + ω + ω^{2} = 0

ω^{3} = 1

→ The product of the n^{th} roots of unity is (-1)^{n-1} .

→ The product of the n^{th} roots of a complex number Z is Z(-1)^{n-1} .

→ ω, ω^{2} are the roots of the equation x^{2} + x + 1 = 0