# De Moivre’s Theorem

De Moivre’s theorem can be simply stated as:

Notice that the sine function on both sides is multiplied by the imaginary unit *i*. This isn’t a pure trigonometrical identity, instead it is a formula that applies to complex numbers.

This formula is an early precursor to Euler’s identity. It is easier to prove — it derives directly from the definition of complex multiplication — but it is not as versatile.

# Complex multiplication

When we first learn about complex multiplication, we simply apply the FOIL rule. For example, if we multiply *1 + i* (which we will call *a*) and *1 + 4i* (which we will call *b*), we get the result *c*, which is *-3 + 5i*:

This is shown here on an Argand diagram:

# Modulus argument form

A complex number can be represented in the form *x + iy*, but it can also be represented in modulus argument form. In that case, we define the complex number using the angle Θ that the number makes with the real axis, and the length *r* from the number to the origin. This diagram shows the two alternative ways of representing the same complex number: