COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS
Expresssin 3θandcos 3θin terms of powers ofcosθandsinθ.Using de Moivre’s theorem,
cos 3θ+isin 3θ=(cosθ+isinθ)^3
=(cos^3 θ−3cosθsin^2 θ)+i(3 sinθcos^2 θ−sin^3 θ). (3.28)We can equate the real and imaginary coefficients separately, i.e.
cos 3θ=cos^3 θ−3cosθsin^2 θ
=4cos^3 θ−3cosθ (3.29)and
sin 3θ=3sinθcos^2 θ−sin^3 θ
=3sinθ−4sin^3 θ.This method can clearly be applied to finding power expansions of cosnθandsinnθfor any positive integern.
The converse process uses the following properties ofz=eiθ,zn+1
zn=2cosnθ, (3.30)zn−1
zn=2isinnθ. (3.31)These equalities follow from simple applications of de Moivre’s theorem, i.e.
zn+1
zn=(cosθ+isinθ)n+(cosθ+isinθ)−n=cosnθ+isinnθ+cos(−nθ)+isin(−nθ)
=cosnθ+isinnθ+cosnθ−isinnθ=2cosnθand
zn−1
zn=(cosθ+isinθ)n−(cosθ+isinθ)−n=cosnθ+isinnθ−cosnθ+isinnθ=2isinnθ.In the particular case wheren=1,
z+1
z=eiθ+e−iθ=2cosθ, (3.32)z−1
z=eiθ−e−iθ=2isinθ. (3.33)