Werner Formulas/Sine by Cosine/Proof 1
Jump to navigation
Jump to search
Theorem
- $\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$
Proof
\(\ds \) | \(\) | \(\ds \frac {\sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} + \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2\) | Sine of Sum and Sine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sin \alpha \cos \beta} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin \alpha \cos \beta\) |
$\blacksquare$