Question #81d1e

1 Answer
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