Question

Verify the identity \sin 3\alpha =3\sin \alpha -4\sin^3\alpha

Collected in the board: Trigonometry

Steven Zheng posted 1 month ago


Answer

\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

Steven Zheng posted 1 month ago

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