Multiple Angle Formulas for Sine and Cosine
Initializing live version
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
Mathematica can directly produce formulas for the sines and cosines of multiple angles.[more]
Here is a derivation of these formulas. Euler's formula is . Then, , but also, . The relation is known as de Moivre's theorem.
For the case , the right-hand side can be expanded to give
using . Equating the imaginary and real parts leads to the double angle formulas
The cases are similar, using a binomial expansion.[less]
Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA