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:
  
<pre><math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math></pre>
+
<math>\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)</math>
  
  

Revision as of 23:35, 9 January 2007

DeMoivre's Theorem is a very useful theorem in the mathematical fields of Complex Numbers. It states that:

$\left(\cos x+i\sin x\right)^n=\cos(nx)+i\sin(nx)$


Proof

This is one proof of DeMoivre's theorem by Mathematical Induction.

If $n>0$

Part 1

For $n=1$, the case is obviously true.

Part 2

Assume true for the case $n=k$.

Part 3

Now, the case of $n=k+1$.

DeMoivreInductionP1.gif

Therefore, the result is true for all positive integers $n$.

If $n=0$

The formula holds true when $n=0$ because $\cos(0x)+i\sin (0x)=1+i0=1$. Since $z^0=1$, the equation holds true.

If $n<0$

If $n<0$, one must consider $n=-m$ when $m$ is a positive integer.

Therefore:

DeMoivreInductionP2.gif

And thus, the formula proves true for all integral values of $n$. $\Box$