Werner Formulas/Cosine by Sine

From ProofWiki
Jump to navigation Jump to search

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

\(\ds \cos \alpha \sin \beta\) \(=\) \(\ds \sin \beta \cos \alpha\)
\(\ds \) \(=\) \(\ds \frac {\map \sin {\beta + \alpha} + \map \sin {\beta - \alpha} } 2\) Werner 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 presented as

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

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


Examples

Example: $2 \cos 30 \degrees \sin 20 \degrees$

$2 \cos 30 \degrees \sin 20 \degrees = \sin 50 \degrees - \sin 10 \degrees$


Example: $2 \cos 2 A \sin 4 A$

$2 \cos 2 A \sin 4 A = \sin 6 A + \sin 2 A$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Werner_Formulas/Cosine_by_Sine&oldid=680500"