Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 2 De Moivre’s Theorem will help students to solve mathematical problems quickly.

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

→ cos θ + i sin θ = e^{iθ}

→ cos θ – i sin θ = e^{-iθ}

→ (cos θ + i sin θ)^{n} = cos nθ + isin nθ = e^{iθ}

If n is an integer (De Moivre’s theorem for integral index)

→ n is a rational number then (cos θ + i sin θ)^{n} = cos nθ + i sin nθ

→ If z_{0} = r_{0}(cos θ + isin θ), then nth roots of z_{0} = z^{n}.

→ The n^{th} root of unity

z^{n} = 1

z = (1)^{1/n}

nth roots of unity are cis \(\frac{2 \mathrm{k} \pi}{\mathrm{n}}\). k = 0, 1, 2………………..(n – 1)

→ Cube roots of unitv:

1, ω, ω^{2}

ω = \(\frac{-1+\mathrm{i} \sqrt{3}}{2}\), ω^{2} = \(\frac{-1-\mathrm{i} \sqrt{3}}{2}\)

1 + ω + ω^{2} = 0

ω^{3} = 1

→ Fourth roots of unity:

z^{4} = 1 or z = (1)^{1/4}

z = 1, -1, +i, -i