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 = eiθ
→ cos θ – i sin θ = e-iθ
→ (cos θ + i sin θ)n = cos nθ + isin nθ = eiθ
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 z0 = r0(cos θ + isin θ), then nth roots of z0 = zn.
→ The nth root of unity
zn = 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, ω2
ω = \(\frac{-1+\mathrm{i} \sqrt{3}}{2}\), ω2 = \(\frac{-1-\mathrm{i} \sqrt{3}}{2}\)
1 + ω + ω2 = 0
ω3 = 1
→ Fourth roots of unity:
z4 = 1 or z = (1)1/4
z = 1, -1, +i, -i