De Moivre's theorem states that for
where
The theorem is easy to prove using the relationship
Raising both sides of this expression to the power of
gives
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 for
and
using 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
Use
to give
Simplifying this expression gives
Equating imaginary coefficients of (1) and (2) gives respectively
Use
to 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