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

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

Also see

• Simpson's Formula for Sine by Sine
• Simpson's Formula for Cosine by Cosine
• Simpson's Formula for Sine by Cosine

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
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Product formulae