What Is de Moivre's Formula and How Do You Use It?

Complex numbers can be used to solve problems that seem at first glance to only deal with real numbers. An important tool you can use in these cases is de Moivre’s formula.

Calculations are often simplified by moving the exponent as in de Moivre’s formula. This is demonstrated in Example 1.

De Moivre’s formula can be proved using Euler’s formula and simple power rules:

Using Euler’s formula, you can derive a relationship between the exponential function and the trigonometric functions:

$$r{e}^{i𝜃}=r\left(\cos 𝜃+i\sin 𝜃\right).$$

You can therefore define cosine and sine using complex numbers via Euler’s formula.

The definition can be justified for real numbers $𝜃\in ℝ$ using Euler’s formula like this:

$$\begin{array}{llll}\hfill \frac{e^{i𝜃}+e^{-i𝜃}}{2}& =\frac{\cos 𝜃+i\sin 𝜃+\cos \left(-𝜃\right)+i\sin \left(-𝜃\right)}{2}& \hfill & \\ \hfill & =\frac{\cos 𝜃+i\sin 𝜃+\cos 𝜃-i\sin 𝜃}{2}& \hfill & \\ \hfill & =\frac{\text{<mn>2</mn>}\cos 𝜃}{\text{<mn>2</mn>}}& \hfill & \\ \hfill & =\cos 𝜃,& \hfill & \end{array}$$

and

$$\begin{array}{llll}\hfill \frac{e^{i𝜃}-e^{-i𝜃}}{2i}& =\frac{\cos 𝜃+i\sin 𝜃-\left(\cos \left(-𝜃\right)+i\sin \left(-𝜃\right)\right)}{2i}& \hfill & \\ \hfill & =\frac{\cos 𝜃+i\sin 𝜃-\cos 𝜃+i\sin 𝜃}{2i}& \hfill & \\ \hfill & =\frac{\text{<mn>2</mn><mi type="i">i</mi>}\sin 𝜃}{\text{<mn>2</mn><mi type="i">i</mi>}}& \hfill & \\ \hfill & =\sin 𝜃.& \hfill & \end{array}$$

The relationship between the exponential function and the trigonometric functions is useful in a variety of situations. It is often easier to work with the exponential function than the trigonometric functions. So when you’re working with trigonometric functions, it can be a good idea to reformulate the problem using the exponential function.