Algebra - Complex numbers — De Moivre's theorem

About this lesson

This unit covers De Moivre's theorem with positive integer powers and with positive rational powers, as well as representing roots of complex numbers in the Argand plane. It begins by stating the theorem for positive integer n: (cos θ + i sin θ)^n = cos nθ + i sin nθ, and uses it with the binomial theorem to express cos 3θ in terms of cos θ. It then extends to rational powers, explaining that (cos θ + i sin θ)^k takes several values depending on an integer r, and shows how to find roots of equations like Z^4 = 8(1 - √3 i) by converting to trigonometric form and applying the theorem. The unit also covers finding square roots of complex numbers algebraically, solving quadratic equations with complex coefficients, and representing roots as points on a circle in the Argand plane, forming regular polygons. By the end, a student should be able to apply De Moivre's theorem to find powers and roots of complex numbers, express roots in trigonometric and exponential forms, solve related equations, and represent roots graphically.