De Moivre's Formula
Pronunciation: /də ˈmwɑvrə ˈfɔrmyələ/ Explain
De Moivre's formula is
.
^{[1]}
This formula follows from the Euler relation,
Applying the power rule for exponents,
gives
Now use the Euler relation to substitute both sides of the equation
Example
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.
References
- Bauer, George N. and Brooke, W. E.. Plane and Spherical Trigonometry. 2nd revised edition. pp 113-125. www.archive.org. D. C. Heath & Co.. 1917. Last Accessed 8/6/2018. http://www.archive.org/stream/planesphericaltr00bauerich#page/113/mode/1up/search/Moivre. Buy the book
- Rothrock, David A.. Elements of plane and spherical trigonometry. pp 101-107. www.archive.org. The Macmillan Company. 1917. Last Accessed 8/6/2018. http://www.archive.org/stream/elementsofplanes00rothiala#page/101/mode/1up/search/Moivre. Buy the book
Cite this article as:
McAdams, David E. De Moivre's Formula. 7/4/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/demoivresformula.html.
Image Credits
Revision History
7/3/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (
McAdams, David E.)
1/22/2010: Added "References". (
McAdams, David E.)
12/3/2008: Initial version. (
McAdams, David E.)