Roots of Unity
Solutions to zⁿ = 1 are equally spaced on the unit circle.
Key idea
Solutions to zⁿ = 1 are equally spaced on the unit circle.
In depth
The nth roots of 1 are cos(2πk/n) + i sin(2πk/n) for k = 0, ..., n − 1. They are evenly spaced on the unit circle, forming a regular n-gon, and their sum is 0.
Worked example
Cube roots of 1: 1, −1/2 + (√3/2)i, −1/2 − (√3/2)i.
Formula
Real-world uses
- Designing the Fast Fourier Transform used in audio, images and mobile communications
Common mistake
Giving only the real root.
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 Roots of Unity?
Solutions to zⁿ = 1 are equally spaced on the unit circle. It is a topic at the expert level in Trigonometry.
What is the formula for Roots of Unity?
The key formula is e^(2πik/n).
Where is Roots of Unity used in real life?
Real-world examples: Designing the Fast Fourier Transform used in audio, images and mobile communications.
What should I learn before Roots of Unity?
Make sure you are comfortable with De Moivre’s Theorem, the previous topic in Trigonometry.
What is a common mistake with Roots of Unity?
Giving only the real root.
How can I practise Roots of Unity?
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 Roots of Unity?
Move on to Euler’s Formula 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).