DeMoivre’s Theorem: A Mathematical Gift That Keeps on Giving
DeMoivre’s Theorem (written above) is a simple theorem relating complex numbers to trigonometry. It was proposed and proved for all positive integers n by Abraham DeMoivre (1667–1754), a French mathematician who was a religious exile in England from a young age and a friend of such greats as Isaac Newton and Edmond Halley.
In 1749, near the end of DeMoivre’s life, Leonhard Euler would prove the result for all real n as a fairly trivial consequence of Euler’s formula. If you have read any math articles on Medium, you will no doubt have read about Euler’s formula, because math writers on Medium are absolutely obsessed with it.
But today I want to give DeMoivre’s Theorem some love, because you can do some pretty amazing things with it. In this article I’m going to show you how you can use it to derive higher order trigonometric identities by using the binomial expansion and matching real or complex parts. I’m also going to show you how this gives a path to finding the precise real roots of certain higher order polynomials. This in turn offers routes to finding closed surd forms of certain trigonometric ratio values.
Deriving higher order trigonometric identities
Let’s see how we can express cos(4θ) in terms of cosθ using DeMoivre’s Theorem. We will start with the theorem statement:
Now let’s expand the left hand side using the Binomial Theorem. But before we do that, we can note that cos(4θ) is the real part of the right hand side, and that should correspond with all even powered terms in our binomial expansion, since even powers of i are real. From this observation, we can derive that:
Using a similar approach we can express cos(6θ) in terms of cosθ:
Finding precise real roots of higher order polynomials
Now consider this sixth-order polynomial equation for which we want to find the real roots: