Werner Formulas/Sine by Cosine

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Theorem

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

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


Proof 1

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

$\blacksquare$


Proof 2

\(\ds \) \(\) \(\ds 2 \sin \alpha \cos \beta\)
\(\ds \) \(=\) \(\ds 2 \paren {\dfrac {\map \exp {i \alpha} - \map \exp {-i \alpha} } {2 i} } \paren {\dfrac {\map \exp {i \beta} + \map \exp {-i \beta} } 2}\) Euler's Sine Identity and Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\map \exp {i \alpha} - \map \exp {-i \alpha} } \paren {\map \exp {i \beta} + \map \exp {-i \beta} }\)
\(\ds \) \(=\) \(\ds \frac 1 {2 i} \paren {\map \exp {i \paren {\alpha + \beta} } - \map \exp {-i \paren {\alpha + \beta} } + \map \exp {i \paren {\alpha - \beta} } - \map \exp {-i \paren {\alpha - \beta} } }\)
\(\ds \) \(=\) \(\ds \frac {\map \exp {i \paren {\alpha + \beta} } - \map \exp {-i \paren {\alpha + \beta} } } {2 i} + \frac {\map \exp {i \paren {\alpha - \beta} } - \map \exp {-i \paren {\alpha - \beta} } } {2 i}\)
\(\ds \) \(=\) \(\ds \map \sin {\alpha + \beta} + \map \sin {\alpha - \beta}\)

$\blacksquare$


Also presented as

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

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


Examples

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

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


Also see


Sources

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