# Simpson's Formulas/Cosine by Sine

## Theorem

$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$

where $\cos$ denotes cosine and $\sin$ denotes sine.

## Proof 1

 $\ds$  $\ds \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ $\ds$ $=$ $\ds \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} - \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2$ Sine of Sum and Sine of Difference $\ds$ $=$ $\ds \frac {2 \cos \alpha \sin \beta} 2$ $\ds$ $=$ $\ds \cos \alpha \sin \beta$

$\blacksquare$

## Proof 2

 $\ds \cos \alpha \sin \beta$ $=$ $\ds \sin \beta \cos \alpha$ $\ds$ $=$ $\ds \frac {\map \sin {\beta + \alpha} + \map \sin {\beta - \alpha} } 2$ Simpson's Formula for Sine by Cosine $\ds$ $=$ $\ds \frac {\map \sin {\alpha + \beta} + \map \sin {-\paren {\alpha - \beta} } } 2$ $\ds$ $=$ $\ds \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$ Sine Function is Odd

$\blacksquare$

## Also reported as

This result can also sometimes be seen as:

$2 \cos \alpha \sin \beta = \map \sin {\alpha + \beta} - \map \sin {\alpha - \beta}$

## Examples

### Example: $2 \cos 30 \degrees \sin 20 \degrees$

$2 \cos 30 \degrees \sin 20 \degrees = \sin 50 \degrees - \sin 10 \degrees$

### Example: $2 \cos 2 A \sin 4 A$

$2 \cos 2 A \sin 4 A = \sin 6 A + \sin 2 A$