Corollary 1.2.15: DeMoivre's Theorem |
For any integer n and any real number t we have
(cos(t) + i sin(t))n = cos(nt) + i sin(nt)
Context
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DeMoivre's Formula is quite something. It says that if you take a number
on the unit circle (i.e. with lenght 1) with initial argument (angle) t
and multiply it by itself, it simply rotates around the unit circle by that
angle t. Each time you multiply the number by itself, the
vector rotates another t degrees. In other words, in this case
the power operator results in a simple rotation.
Powers of a vector z with |z|=1
|
The proof of the formula follows directly from our previous theorem on
multiplying directly. We showed that if z = r cis(t)
then zn = rn cis(nt). Now let
r=1 and substitute the definition of cis to get
DeMoivre's Formula.
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Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007