De-Moivre’s Theorem and Euler Formulas

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List of De-Moivre’s Theorem and Euler Formulas

General De-Moivre’s Theorem and Euler Formulas are stated below and you can make the most out of them. Learn the concept easily and overcome the hectic task of calculations by referring to the formulae over here. Quickly grasp them and do it the right way while solving your problems.

1. De- Moiver’s Theorem:

It states that if n is rational number (positive, negative or zero) then
(cos θ + i sin θ)n = cos nθ + i sin nθ &
(cos θ + i sin θ)-n = cos nθ – i sin nθ ; n ∈ Q

2. Euler’s Formula

eiθ = cos θ + i sin θ and e-iθ = cos θ – i sin θ
eiθ + e-iθ = 2 cos θ and eiθ – e-iθ = 2i sin θ

3. nth roots of complex number (z1/n)

= r1/n \(\left[\cos \left(\frac{2 m \pi+\theta}{n}\right)+i \sin \left(\frac{2 m \pi+\theta}{n}\right)\right]\) where m = 0, 1, 2, …….., (n – 1)

  • Sum of all roots of z1/n is always equal to zero
  • Product of all roots of z1/n = (-1)n-1 z

4. Cube root of unity

1 + ω + ω2 = 0, ω3 = 1 where ω = – \(\frac{1}{2}+\frac{i \sqrt{3}}{2}\)

5. Continued product of the roots

  • Continued product of roots of z1/n = \(\left\{\begin{array}{l}\mathrm{z}, \text { if } \mathrm{n} \text { is odd } \\-\mathrm{z}, \text { if } \mathrm{n} \text { is even }\end{array}\right.\)
  • Continued product of values of zm/n = \(\left\{\begin{array}{c}z^{m}, \text { if } n \text { is odd } \\(-z)^{m}, \text { if } n \text { is even }\end{array}\right.\)

6. The sum of pth powers of nth roots of unity

\(\begin{equation}=\left\{\begin{array}{l}n \text { when pis a multiple of } n \\0 \text { when } p \text { is not a multiple of } n\end{array}\right.\end{equation}\)

7. Some important results

If z = cos θ + i sin θ

  • z + \(\frac{1}{z}\) = 2 cos θ
  • z – \(\frac{1}{z}\) = 2 i sin θ
  • zn + \(\frac{1}{z^{n}}\) =2 cos nθ, zn – \(\frac{1}{z^{n}}\) = 2i sin nθ
  • If x = cos α + i sin α , y = cos β + i sin β, z = cos γ + i sin γ and given, x + y + z = 0, then
    • \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\) = 0
    • yz + zx + xy = 0
    • x2 + y2 + z2 = 0
    • x3 + y3 + z3 = 3xyz

Hope our collection of De-Moivre’s Theorem and Euler Formulas helped you solve complex numbers problems easily. If you need any assistance on other concepts and their related formulas do visit Onlinecalculator.guru a reliable source.