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 Context

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