Difference between revisions of "De Moivre's Theorem"
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DeMoivre's Theorem is a very useful theorem in the mathematical fields of [[Complex Numbers]]. It states that: | DeMoivre's Theorem is a very useful theorem in the mathematical fields of [[Complex Numbers]]. It states that: | ||
− | + | <math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math> | |
Revision as of 22:35, 9 January 2007
DeMoivre's Theorem is a very useful theorem in the mathematical fields of Complex Numbers. It states that:
Contents
[hide]Proof
This is one proof of DeMoivre's theorem by Mathematical Induction.
If
Part 1
For , the case is obviously true.
Part 2
Assume true for the case .
Part 3
Now, the case of .
Therefore, the result is true for all positive integers .
If
The formula holds true when because . Since , the equation holds true.
If
If , one must consider when is a positive integer.
Therefore:
And thus, the formula proves true for all integral values of .