Any pair
a, b
of real numbers can be represented in polar
coordinates with
a = r cosθ and
b = r sinθ where
r =
is the distance from the point
a, b
to the
origin and θ is an angle satisfying
tanθ =
. Thus
any complex number can be written in the form
DeMoivre's Theorem says that if z = rcosθ + i sinθ
and n is a positive integer, then
With Scientific Notebook, you can obtain this result for small values of n by the sequence of operations Expand followed by MenuDialogCombine + Trig Functions and then Factor.
Expand, Combine + Trig Functions, Factor
r
cos t + i sin t
= r3cos3t + 3ir3cos2t sin t - 3r3cos t sin2t - ir3sin3t
= r3cos 3t + ir3sin 3t = r3cos 3t + i sin 3t
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Or, you can use Simplify followed by Combine + Trig Functions and Factor.
Simplify, Combine + Trig Functions, Factor
r
cos t + i sin t
= 4r3cos3t + 4ir3cos2t sin t - 3r3cos t - ir3sin t
= r3cos 3t + ir3sin 3t = r3cos 3t + i sin 3t
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You can get the same results in complete generality by working with reit, since
You can get the identity
Evaluate, Factor
reit = r cos t + ir sin t = rcos t + i sin t
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