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Simple Multiple Angle Formula Solutions using DeMoivre's Theorem and Euler's Relation

The double angle formulae are easy to learn:

Sin 2x = 2 sin x cos x
Cos 2x = (cos x)^2 - (sin x)^2

but working out Cos 4x say in terms of cos x and sin x by using identities such as
Cos(A + B) = Cos A Cos B - Sin A Sin B is laborious.

Find a simple method to work out any multiple angle formula, using De Moivre's Theorem and Euler's relation.

Solution Preview

De Moivre's Theorem states that (Exp ix)^n = Exp (inx) where i is the imaginary number ie i^2 = -1.

We can combine this with Euler's relation Exp iy = Cos y + i Sin y to obtain
(Exp ix)^n = Cos nx + i Sin nx.

The Cos nx part is known as the "real" part and the Sin nx as the "imaginary" ...

Solution Summary

Multiple Angle Formula solutions are found in a simple fashion using DeMoivre's Theorem and Euler's Relation.