2.5.1—de Moivre's theorem

If z=r(cosθ+i sinθ), then z^n=r^n(cos nθ+i sin nθ). The theorem also gives a systematic route to trigonometric identities and roots of complex numbers.

Convert to modulus–argument form, multiply the argument by n and convert back only at the end. Arguments are defined modulo 2π, so equivalent angles represent the same complex number.

(cosθ+i sinθ)^3=cos3θ+i sin3θ. Equating real parts gives cos3θ=4cos³θ−3cosθ.

The modulus is raised to n as well as the angle; multiplying only the angle gives the wrong magnitude.