Difference between revisions of "De Moivre's Theorem"
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:Assume true for the case <math>n=k</math>. Now, the case of <math>n=k+1</math>: | :Assume true for the case <math>n=k</math>. Now, the case of <math>n=k+1</math>: | ||
− | + | \begin{align*} | |
(\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \\ | (\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \\ | ||
& =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by the Assumption in Step II } \\ | & =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by the Assumption in Step II } \\ | ||
& =\cos (k x) \cos x-\sin (k x)+i[\cos (k x) \sin x+\sin (k x) \cos x] & \\ | & =\cos (k x) \cos x-\sin (k x)+i[\cos (k x) \sin x+\sin (k x) \cos x] & \\ | ||
& =\operatorname{cis}(k+1) & \text { Various Trigonometric Identities } | & =\operatorname{cis}(k+1) & \text { Various Trigonometric Identities } | ||
− | \end{align} | + | \end{align*} |
:Therefore, the result is true for all positive integers <math>n</math>. | :Therefore, the result is true for all positive integers <math>n</math>. |
Revision as of 03:03, 6 February 2022
DeMoivre's Theorem is a very useful theorem in the mathematical fields of complex numbers. It allows complex numbers in polar form to be easily raised to certain powers. It states that for and
,
.
Proof
This is one proof of De Moivre's theorem by induction.
- If
, for
, the case is obviously true.
- Assume true for the case
. Now, the case of
:
\begin{align*} (\cos x+i \sin x)^{k+1} & =(\cos x+i \sin x)^{k}(\cos x+i \sin x) & \text { by Exponential laws } \\ & =[\cos (k x)+i \sin (k x)](\cos x+i \sin x) & \text { by the Assumption in Step II } \\ & =\cos (k x) \cos x-\sin (k x)+i[\cos (k x) \sin x+\sin (k x) \cos x] & \\ & =\operatorname{cis}(k+1) & \text { Various Trigonometric Identities } \end{align*}
- Therefore, the result is true for all positive integers
.
- If
, the formula holds true because
. Since
, the equation holds true.
- If
, one must consider
when
is a positive integer.
And thus, the formula proves true for all integral values of .
Note that from the functional equation where
, we see that
behaves like an exponential function. Indeed, Euler's identity states that
. This extends De Moivre's theorem to all
.