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Rewrite \(\sin 3 \alpha\) in terms of \(\sin \alpha\) :


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Rewrite \(\sin 3 \alpha\) in terms of \(\sin \alpha\) :
in Mathematics by ♦ Diamond (71,587 points) | 8 views

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\(\sin 3 \alpha=\sin (2 \alpha+\alpha)\)
\(=\sin 2 \alpha \cdot \cos \alpha+\cos 2 \alpha \cdot \sin \alpha\)
\(=(2 \sin \alpha \cdot \cos \alpha) \cdot \cos \alpha+\left(1-2 \sin ^2 \alpha\right) \cdot \sin \alpha\)
\(=2 \sin \alpha \cdot \cos ^2 \alpha+\sin \alpha-2 \sin ^3 \alpha\)
\(=2 \sin \alpha \cdot\left(1-\sin ^2 \alpha\right)+\sin \alpha-2 \sin ^3 \alpha\)
\(=2 \sin \alpha-2 \sin ^3 \alpha+\sin \alpha-2 \sin ^3 \alpha\)
\(=3 \sin \alpha-4 \sin ^3 \alpha\)
by ♦ Diamond (71,587 points)

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