de Moivre's theorem
(cosθ + i sinθ)n = cos nθ + i sin nθ. Derive multiple-angle identities.
Powers of complex numbers in polar form: raise modulus to the power, multiply argument by the power. Used to derive sin(nθ), cos(nθ) identities and find nth roots.
Worked examples
(1 + i)4: |1+i|=√2, arg=π/4. Result: (√2)4(cos π + i sin π) = 4·(−1) = −4.
cos 3θ = 4cos³θ − 3cosθ (from de Moivre with n = 3).
nth roots of 1: e^(2πik/n) for k = 0, 1, ..., n−1.
Frequently asked questions
Why use polar?
Powers and roots are trivial in polar form. Hard in Cartesian.
Negative or fractional n?
Works the same way; gives 1/zn or roots.