De Moivre's formula (or) De Moivre's theorem is related to complex numbers. We can expand the power of a complex number just like how we expand the power of any binomial. But De Moivre's formula simplifies the process of finding the power of a complex number much simple. To apply De Moivre's formula, the complex number first needs to be converted into polar form. Let us learn more about this formula in the upcoming sections.
What Is De Moivre's Formula?
To understand De Moivre's formula, let us consider a complex number in polar form z = r (cos θ + i sin θ). Let us raise this to powers 2 and see what happens.
Similarly, we can see that (r (cos θ + i sin θ))3 = r3 (cos 3θ + i sin 3θ) by expanding it manually. This is the basic idea of De Moivre formula. If a complex number (in polar form) r (cos θ + i sin θ) is raised to some power 'n' (where n is an integer), then the modulus of the result is rn and the argument of the result is nθ. Thus, De Moivre Formula is:
(r (cos θ + i sin θ))n = rn (cos nθ + i sin nθ), where n ∈ Z