De Moivre‘s theorem states that forwhere

The theorem is easy to prove using the relationship Raising both sides of this expression to the power ofgives

The theorem is useful when deriving relationships between trigonometric functions. For example, we can obtain polynomial expressions for sin n %theta and cos n %theta for any n using de Moivre’s theorem.

Example: Derive expressions forandusing de Moivre‘s theorem.

(1)

Expanding the left hand side using the binomial theorem gives

(2)

Equating real coefficients of (1) and (2) gives respectively

Useto give Simplifying this expression gives

Equating imaginary coefficients of (1) and (2) gives respectively

Useto give Simplifying this expression gives

(1)

Expanding the left hand side using the binomial theorem gives

The real and imaginary parts of this expression are

Equating real parts gives

Useto give

This simplifies to

Equating imaginary parts gives

Useto give

This simplifies to