De Moivre's Formula

Pronunciation: /də ˈmwɑvrə ˈfɔrmyələ/ ?

De Moivre's formula is

This formula follows from the Euler relation,
Applying the power rule for exponents,
Now use the Euler relation to substitute both sides of the equation
(cos(theta)+i*sin(theta))^n = (cos(r*theta)+i*sin(r*theta)


Some of the trigonometric identities can be derived using De Moivre's Formula. For double angle formulas, start with the expression

Apply De Moivre's formula, giving
Now expand the right side of the equation using the binomial theorem.
Since the imaginary parts of both sides of the equations must be equal and the real parts of both sides of the equations must be equal, this gives two identities.
cos(2*theta)=cos(theta)^2-sin(theta)^2, sin(2*theta)=2*cos(theta)*sin(theta)


  1. Bauer, George N. and Brooke, W. E.. Plane and Spherical Trigonometry, 2nd revised edition, pp 113-125. D. C. Heath & Co., 1917. (Accessed: 2010-01-22).
  2. Rothrock, David A.. Elements of plane and spherical trigonometry, pp 101-107. The Macmillan Company, 1917. (Accessed: 2010-01-22).

Printed Resources

Cite this article as:

De Moivre's Formula. 2010-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC.


Image Credits

Revision History

2010-01-22: Added "References" (McAdams, David.)
2008-12-03: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License