Multiple Angle Formulas for Sine and Cosine

n

2

3

4

5

6

7

8

9

10

sin(2θ) = 2sin(θ)cos(θ)

cos(2θ) =

2

cos

(θ)-

2

sin

(θ)

Mathematica can directly produce formulas for the sines and cosines of multiple angles.

Here is a derivation of these formulas. Euler's formula is

iθ

e

=cosθ+isinθ

. Then,

inθ

e

=cos(nθ)+isin(nθ)

, but also,

inθ

e

=

n

(cosθ+isinθ)

. The relation

cos(nθ)+isin(nθ)=

n

(cosθ+isinθ)

is known as de Moivre's theorem.

For the case

n=2

, the right-hand side can be expanded to give

cos(2θ)+isin(2θ)=

2

cos

θ+2icosθsinθ-

2

sin

θ

,

using

2

i

=-1

. Equating the imaginary and real parts leads to the double angle formulas

sin(2θ)=2cosθsinθ

,

cos(2θ)=

2

cos

θ-

2

sin

θ

.

The cases

n=3,4,5,…

are similar, using a binomial expansion.