Multiple Angle Formulas for Sine and Cosine

Initializing live version
Download to Desktop

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.


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.


Contributed by: S. M. Blinder (July 2018)
Open content licensed under CC BY-NC-SA



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.