De Moivre's Formula

Pronunciation: /də ˈmwɑvrə ˈfɔrmyələ/ ?

De Moivre's formula is

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(theta)+i*sin(theta))^n = (cos(r*theta)+i*sin(r*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. 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). http://www.archive.org/stream/planesphericaltr00bauerich#page/113/mode/1up/search/Moivre.
  2. Rothrock, David A.. Elements of plane and spherical trigonometry, pp 101-107. The Macmillan Company, 1917. (Accessed: 2010-01-22). http://www.archive.org/stream/elementsofplanes00rothiala#page/101/mode/1up/search/Moivre.

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De Moivre's Formula. 2010-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/demoivresformula.html.

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

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