\sin 3\alpha = \sin (\alpha +2\alpha )

=\sin \alpha \cos 2\alpha +\cos \alpha \sin 2\alpha

=\sin \alpha (1-2\sin^2\alpha )+2\sin \alpha \cos^2\alpha

=\sin \alpha - 2\sin^3\alpha +2\sin \alpha (1-\sin^2\alpha )

=3\sin \alpha-4\sin^3\alpha