Question #81d1e

1 Answer
Eddie
May 4, 2017

By deMoivre:

#color(red)((cos 3 theta + i sin 3 theta) )= (cos theta + i sin theta)^3#

And the RHS has binomial expansion:

#(cos theta + i sin theta)^3 = color(red)( cos^3 theta + 3 i cos^2 theta sin theta - 3 cos theta sin^2 theta - i sin^2 theta )#

Collecting the real parts of the equivalent red expressions:

#cos 3 theta = cos^3 theta - 3 cos theta sin^2 theta #

#= cos^3 theta - 3 cos theta (1 - cos^2 theta) #

#implies cos 3 theta = 4 cos^3 theta - 3 cos theta # and #A = 4, B = -3#

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