Werner Formulas/Cosine by Cosine
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Theorem
- $\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$
where $\cos$ denotes cosine.
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 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {2 \cos \alpha \cos \beta} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \alpha \cos \beta\) |
$\blacksquare$
Also presented as
The Werner Formula for Cosine by Cosine can also be seen in the form:
- $2 \cos \alpha \cos \beta = \map \cos {\alpha - \beta} + \map \cos {\alpha + \beta}$
Examples
Example: $2 \cos 20 \degrees \cos 50 \degrees$
- $2 \cos 20 \degrees \cos 50 \degrees = \cos 30 \degrees + \cos 70 \degrees$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.66$
- 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