Werner Formulas
Theorem
Cosine by Cosine
- $\cos \alpha \cos \beta = \dfrac {\map \cos {\alpha - \beta} + \map \cos {\alpha + \beta} } 2$
Sine by Sine
- $\sin \alpha \sin \beta = \dfrac {\map \cos {\alpha - \beta} - \map \cos {\alpha + \beta} } 2$
Sine by Cosine
- $\sin \alpha \cos \beta = \dfrac {\map \sin {\alpha + \beta} + \map \sin {\alpha - \beta} } 2$
Cosine by Sine
- $\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$
Hyperbolic Functions
While Johann Werner did not consider the hyperbolic functions, it is convenient to use his name to identify them, as follows:
Hyperbolic Cosine by Hyperbolic Cosine
- $\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$
Hyperbolic Sine by Hyperbolic Sine
- $\sinh x \sinh y = \dfrac {\map \cosh {x + y} - \map \cosh {x - y} } 2$
Hyperbolic Sine by Hyperbolic Cosine
- $\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$
Also presented as
The Werner Formulas can also be presented as follows:
Sine by Sine
The Werner Formula for Sine by Sine can also be seen in the form:
- $2 \sin \alpha \sin \beta = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}$
Sine by Cosine
The Werner Formula for Sine by Cosine can also be seen in the form:
- $2 \sin \alpha \cos \beta = \map \sin {\alpha - \beta} + \map \sin {\alpha + \beta}$
Cosine by Sine
The Werner Formula for Cosine by Sine can also be seen in the form:
- $2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$
Cosine by Cosine
The Werner Formula for Cosine by Cosine can also be seen in the form:
- $2 \cos \alpha \cos \beta = \map \cos {\alpha - \beta} + \map \cos {\alpha + \beta}$
Also known as
The Werner Formulas are also known as the Product Formulas or Product Formulae.
They can also be called the Product-to-Sum Formulas.
Some sources call them Simpson's Formulas, but this appears not to be backed up by the literature, and this name is usually applied to another set of formulas altogether.
Also see
Source of Name
This entry was named for Johann Werner.
Sources
- 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