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

nth roots of unity:
→ nth roots of unity are {1, ω, ω2 …….. ωn – 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 nth root of unity, then

  • ωn = 1
  • 1 + ω + ω2 + ………… + ωn – 1 = 0

Inter 2nd Year Maths 2A De Moivre’s Theorem Formulas

Cube roots of unity:
→ 1, ω, ω2 are cube roots of unity when

  • ω3 = 1
  • 1 + ω + ω2 = 0
  • ω = \(\frac{-1+i \sqrt{3}}{2}\), ω2 = \(\frac{-1-i \sqrt{3}}{2}\)
  • Fourth roots of unity roots are 1, – 1, i, – i

→ If Z0 = r0 cis θ0 ≠ 0, then the nth roots of Z0 are αk = (r0)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

→ xn + \(\frac{1}{x^{n}}\) = 2cos nθ, xn – \(\frac{1}{x^{n}}\) = 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 nth roots of a complex number in the Argand diagram are concyclic.

→ The points representing nth 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, w2,………. , wn-1 where w = cis\(\frac{2 \pi}{n}\)

Inter 2nd Year Maths 2A De Moivre’s Theorem Formulas

→ 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, ω2 where ω = cis\(\frac{2 \pi}{3}\), ω2 = cis\(\frac{4 \pi}{3}\) or
ω = \(\frac{-1+i \sqrt{3}}{2}\)
ω2 = \(\frac{-1-i \sqrt{3}}{2}\)
1 + ω + ω2 = 0
ω3 = 1

→ The product of the nth roots of unity is (-1)n-1 .

→ The product of the nth roots of a complex number Z is Z(-1)n-1 .

→ ω, ω2 are the roots of the equation x2 + x + 1 = 0