Multiple Angle Formulas for Sine and Cosine
Multiple Angle Formulas for Sine and Cosine
Mathematica can directly produce formulas for the sines and cosines of multiple angles.
Here is a derivation of these formulas. Euler's formula is =cosθ+isinθ. Then, =cos(nθ)+isin(nθ), but also, =. The relation is known as de Moivre's theorem.
iθ
e
inθ
e
inθ
e
n
(cosθ+isinθ)
cos(nθ)+isin(nθ)=
n
(cosθ+isinθ)
For the case , the right-hand side can be expanded to give
n=2
cos(2θ)+isin(2θ)=θ+2icosθsinθ-θ
2
cos
2
sin
using =-1. Equating the imaginary and real parts leads to the double angle formulas
2
i
sin(2θ)=2cosθsinθ
cos(2θ)=θ-θ
2
cos
2
sin
The cases are similar, using a binomial expansion.
n=3,4,5,…