De Moivre's Formula

Pronunciation: /də ˈmwɑvrə ˈfɔɹ.mjə.lə/ Explain

De Moivre's formula is

(cos(theta)+i*sin(theta))^n = (cos(r*theta)+i*sin(r*theta).[2]
This formula follows from the Euler relation,
e^(i*theta)=cos(theta)+i*sin(theta)
Applying the power rule for exponents,
(a^n)^m=a^(n*m)
gives
(e^(i*theta))^n=e^(i*n*theta)
Now use the Euler relation to substitute both sides of the equation
cos(2*theta)+i*sin(2*theta)

Example

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

cos(2*theta)+i*sin(2*theta)
Apply De Moivre's formula, giving
cos(2*theta)+i*sin(2*theta)=(cos(theta)+i*sin(theta))^2
Now expand the right side of the equation using the binomial theorem.
cos(2*theta)+i*sin(2*theta)=cos(theta)^2+2*i*cos(theta)*sin(theta)-sin(theta)^2
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)

References

  1. McAdams, David E.. All Math Words Dictionary, de Moivre's formula. 2nd Classroom edition 20150108-4799968. pg 57. Life is a Story Problem LLC. January 8, 2015. Buy the book
  2. 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 7/3/2018. http://www.archive.org/stream/planesphericaltr00bauerich#page/113/mode/1up/search/Moivre. Buy the book
  3. Rothrock, David A.. Elements of plane and spherical trigonometry. pp 101-107. www.archive.org. The Macmillan Company. 1917. Last Accessed 7/3/2018. 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. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/d/demoivresformula.html.

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Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
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.)

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