# 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.