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…