Question

Expand θ 6 sin in a series of cosines of multiples o θ​

Answer 1

Answer:

Let, x=cosθ+isinθ1x=cosθ−isinθ∴xn=(cosθ+isinθ)n=cosnθ+isinnθAlso1xn=(cosθ−isinθ)n=cosnθ−isinnθ{∵DeMoivre′sTheorem}xn−1xn=2isinnθ…(i)sin7θ=(sinθ)7=[12i(x−1x)]7from(i)=1128×i7(x−1x)7=1128i7[x7−7x6.1x+21x5.1x2−35x4.1x3+35x3.1x4−21x2.1x5+7x.1x6−1x7]…Expandingbinomially=1128i7[(x7−1x7)−7(x5−1x5)+21(x3−1x3)−35(x−1x)]=1128×i7[2isin7θ−7(2isin5θ)+21(2isin3θ)−35(2isinθ)]from(i)=164×i6[sin7θ−7sin5θ+21sin3θ−35sinθ]∴sin7θ=−164[sin7θ−7sin5θ+21sin3θ−35sinθ]