## De Moivre's Theorem

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 Use to give This simplifies to Equating imaginary parts gives Use to give This simplifies to #### Add comment Refresh