Werner Formulas/Sine by Sine

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Theorem

$\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$

where $\sin$ denotes sine and $\cos$ denotes cosine.


Corollary

$\map \sin {A + B} \map \sin {A - B} = \paren {\sin A + \sin B} \paren {\sin A - \sin B}$


Proof

\(\ds \) \(\) \(\ds \frac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2\)
\(\ds \) \(=\) \(\ds \frac {\paren {\cos \alpha \cos \beta + \sin \alpha \sin \beta} - \paren {\cos \alpha \cos \beta - \sin \alpha \sin \beta} } 2\) Cosine of Difference and Cosine of Sum
\(\ds \) \(=\) \(\ds \frac {2 \sin \alpha \sin \beta} 2\)
\(\ds \) \(=\) \(\ds \sin \alpha \sin \beta\)

$\blacksquare$


Also presented as

The Werner Formula for Sine by Sine can also be seen in the form:

$2 \sin \alpha \sin \beta = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}$


Examples

Example: $2 \sin 10 \degrees \sin 30 \degrees$

$2 \sin 10 \degrees \sin 30 \degrees = \cos 20 \degrees - \cos 40 \degrees$


Example: $2 \sin 3 A \sin 5 A$

$2 \sin 3 A \sin 5 A = \cos 2 A - \cos 8 A$


Also see


Sources

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