Edexcel A Level Further Maths: Core Pure:复习笔记1.2.2 de Moivre's Theorem

De Moivre's Theorem

What is de Moivre’s Theorem?

  • de Moivre’s theorem can be used to find powers of complex numbers
  • It states that for
    • Where
      • z ≠ 0
      • r is the modulus, |z|, r ∈ ℝ+
      • θ  is the argument, arg z, θ ∈ ℝ
      • n ∈ ℝ
  • In Euler’s form this is simply:
  • In words de Moivre’s theorem tells us to raise the modulus by the power of n and multiply the argument by n
  • In the formula booklet de Moivre’s theorem is given in both polar and Euler’s form:

How do I use de Moivre’s Theorem to raise a complex number to a power?

  • If a complex number is in Cartesian form you will need to convert it to either modulus-argument (polar) form or exponential (Euler’s) form first
    • This allows de Moivre’s theorem to be used on the complex number
  • You may need to convert it back to Cartesian form afterwards
  • If a complex number is in the form  then you will need to rewrite it as before applying de Moivre’s theorem
  • A useful case of de Moivre’s theorem allows us to easily find the reciprocal of a complex number:
    • Using the trig identities cos(-θ) = cos(θ) and sin(-θ) = - sin(θ) gives
  • In general

Exam Tip

  • You may be asked to find all the powers of a complex number, this means there will be a repeating pattern
    • This can happen if the modulus of the complex number is 1
    • Keep an eye on your answers and look for the point at which they begin to repeat themselves

Worked Example

Find the value of ,  giving your answer in the form a + bi.