De Moivre’s Theorem
Powers of complex numbers.
Key idea
Powers of complex numbers.
In depth
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Raising to a power multiplies the angle by n and raises the modulus to the nth power. It gives quick powers and roots of complex numbers.
Worked example
(1 + i)⁸ = (√2)⁸ (cos 360° + i sin 360°) = 16.
Formula
Real-world uses
- Computing powers of complex numbers in signal processing and quantum physics
Common mistake
Forgetting to raise the modulus to the power.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is De Moivre’s Theorem?
Powers of complex numbers. It is a topic at the expert level in Trigonometry.
What is the formula for De Moivre’s Theorem?
The key formula is zⁿ = rⁿ(cos nθ + i sin nθ).
Where is De Moivre’s Theorem used in real life?
Real-world examples: Computing powers of complex numbers in signal processing and quantum physics.
What should I learn before De Moivre’s Theorem?
Make sure you are comfortable with Complex Numbers in Polar Form, the previous topic in Trigonometry.
What is a common mistake with De Moivre’s Theorem?
Forgetting to raise the modulus to the power.
How can I practise De Moivre’s Theorem?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after De Moivre’s Theorem?
Move on to Roots of Unity next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).