WOLFRAM|DEMONSTRATIONS PROJECT

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
.