From Euler's formula, substituting in
By combining these equations, we can form alternative equations for sine and cosine:
This can be generalised; if we let
A complex number can be raised to a power both using the binomial expansion and de Moivre's Theorem. These two results can then be equated and the real and imaginary components used to find multiple angle formulae for sine and cosine.
Let
Using de Moivre's Theorem:
Equating real and imaginary parts:
The general exponential form of trigonometric functions can be expanded and collected in order to find powers of trigonometric functions in terms of multiple angle functions.
Let
A similar method can be used by expanding
Some sums of trigonometric series can be calculated using the sum of the geometric series formed by the exponential form of the trigonometric functions (if one is formed).
Show that
Consider the geometric series
We can sum the first 10 terms to get
As