Simpson's Formulas/Cosine by Sine

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Theorem

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

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


Proof 1

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

$\blacksquare$


Proof 2

\(\ds \cos \alpha \sin \beta\) \(=\) \(\ds \sin \beta \cos \alpha\)
\(\ds \) \(=\) \(\ds \frac {\map \sin {\beta + \alpha} + \map \sin {\beta - \alpha} } 2\) Simpson's Formula for Sine by Cosine
\(\ds \) \(=\) \(\ds \frac {\map \sin {\alpha + \beta} + \map \sin {-\paren {\alpha - \beta} } } 2\)
\(\ds \) \(=\) \(\ds \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2\) Sine Function is Odd

$\blacksquare$


Also reported as

This result can also sometimes be seen as:

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


Sources