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
- Werner Formula for Cosine by Cosine
- Werner Formula for Sine by Cosine
- Werner Formula for Cosine by Sine
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.65$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): product formulae
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): product formulae
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Product formulae
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Product formulae