Simpson's 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 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 known as

Simpson's formulas are also known as the product formulas or product formulae.

Also see

• Prosthaphaeresis Formulas

Source of Name

This entry was named for Thomas Simpson.

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