De Moivre's Theorem for Trig Identities

n

2

function

sin

cos

sin(2x)	=	Im 2 exp (x)
	=	Im 2 (cos(x)+sin(x)) 
	=	2sin(x)cos(x)

De Moivre's theorem, along with the binomial theorem, can be used to expand functions like

cos(nx)

or

sin(nx)

, where

n

is an integer, into a sum of powers of trig functions consisting of a mixture of sines and cosines. In some cases it is possible to rewrite the expansion such that it contains all sines or all cosines by making use of the identity

2

sin

(x)+

2

cos

(x)=1

.