Complex numbers (Further Pure 2)
De Moivre's theorem
- A complex number in polar form: $z = r(\cos\theta + i\sin\theta)$.
- De Moivre's theorem: for any integer $n$,
$$(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$$
- Use it to expand $\cos n\theta$, $\sin n\theta$ in powers, and to find the $n$ roots of unity ($z^n = 1$, equally spaced on the unit circle).
Practice
De Moivre's theorem states (cos θ + i sin θ)ⁿ equals:
Raising to the power n multiplies the argument by n: cos nθ + i sin nθ.
Practice
How many distinct roots of unity does the equation z⁵ = 1 have?
zⁿ = 1 has exactly n roots, equally spaced on the unit circle; here n = 5.
Practice
The n roots of unity are equally spaced around the unit circle.
They sit at equal angular gaps of 360°/n around the circle.
Example
- Take the real part of $(\cos\theta + i\sin\theta)^3$: $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$.
You've got it
Key idea
- de Moivre: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$
- expands multiple angles like $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$
- the $n$ roots of unity are equally spaced around the unit circle