De Moivre's Theorem
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>
Proof
This is one proof of DeMoivre's theorem by Mathematical Induction.
If ![$n>0$](//latex.artofproblemsolving.com/0/2/4/0247db3d702b5ce6be98f50771750a2723cddcea.png)
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 ![$n=0$](//latex.artofproblemsolving.com/0/2/3/02349576cee613512cdf301a365f06c0760acab5.png)
The formula holds true when because
. Since
, the equation holds true.
If ![$n<0$](//latex.artofproblemsolving.com/7/9/8/79813fe3de65c5970b8c1b6eecc359c0042447ae.png)
If , one must consider
when
is a positive integer.
Therefore:
And thus, the formula proves true for all integral values of .