Question

Express sin 5 theta upon 2 in terms of Cos 5 theta

Answer 1

sin5θ

=sin(3θ+2θ)

=sin3θcos2θ+cos3θsin2θ

=(3sinθ−4sin3θ)(1−2sin2θ)+(4cos3θ−3cosθ)2sinθcosθ

=(3sinθ−4sin3θ−6sin3θ+8sin5θ)+(8cos4θ−6cos2θ)sinθ

=(3sinθ−10sin3θ+8sin5θ)+(8(1−sin2θ)2−6(1−sin2θ))sinθ

=3sinθ−10sin3θ+8sin5θ+(8sin4θ−16sin2θ+8−6+6sin2θ)sinθ

=3sinθ−10sin3θ+8sin5θ+(8sin4θ−10sin2θ+2)sinθ

=3sinθ−4sin3θ−6sin3θ+8sin5θ+8sin5θ−10sin3θ+2sinθ

=16sin5θ−20sin3θ+5sinθ