Prosthaphaeresis formulas

The Prosthaphaeresis formulas convert sums of sines or cosines to products of them:

$\sin A+\sin B$ $=$ $2\sin \left({\displaystyle \frac{A+B}{2}}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)$

$\sin A-\sin B$ $=$ $2\sin \left({\displaystyle \frac{A-B}{2}}\right)\cos \left({\displaystyle \frac{A+B}{2}}\right)$

$\cos A+\cos B$ $=$ $2\cos \left({\displaystyle \frac{A+B}{2}}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)$

$\cos A-\cos B$ $=$ $-2\sin \left({\displaystyle \frac{A+B}{2}}\right)\sin \left({\displaystyle \frac{A-B}{2}}\right)$

We prove the first two using the sine of a sum and sine of a difference formulas:

$\sin (X+Y)$ $=$ $\sin X\cos Y+\cos X\sin Y$

$\sin (X-Y)$ $=$ $\sin X\cos Y-\cos X\sin Y$

Adding or subtracting the two equations yields

$\sin (X+Y)+\sin (X-Y)$ $=$ $2\sin X\cos Y$

$\sin (X+Y)-\sin (X-Y)$ $=$ $2\sin Y\cos X$

If we let $X=\frac{A+B}{2}$ and $Y=\frac{A-B}{2}$, then $X+Y=\frac{2A}{2}=A$ and $X-Y=\frac{2B}{2}=B$, and the last two equations become

$\sin A+\sin B$ $=$ $2\sin \left({\displaystyle \frac{A+B}{2}}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)$

$\sin A-\sin B$ $=$ $2\sin \left({\displaystyle \frac{A-B}{2}}\right)\cos \left({\displaystyle \frac{A+B}{2}}\right)$

as desired.

The last two can be proven similarly, this time using the cosine of a sum and cosine of a difference formulas:

$\cos (X+Y)$ $=$ $\cos X\cos Y-\sin X\sin Y$

$\cos (X-Y)$ $=$ $\cos X\cos Y+\sin X\sin Y$

Adding or subtracting the two equations yields

$\cos (X+Y)+\cos (X-Y)$ $=$ $2\cos X\cos Y$

$\cos (X+Y)-\cos (X-Y)$ $=$ $-2\sin Y\sin X$

Again, if we let $X=\frac{A+B}{2}$ and $Y=\frac{A-B}{2}$, then $X+Y=\frac{2A}{2}=A$ and $X-Y=\frac{2B}{2}=B$, and the last two equations become

$\cos A+\cos B$ $=$ $2\cos \left({\displaystyle \frac{A+B}{2}}\right)\cos \left({\displaystyle \frac{A-B}{2}}\right)$

$\cos A-\cos B$ $=$ $-2\sin \left({\displaystyle \frac{A-B}{2}}\right)\sin \left({\displaystyle \frac{A+B}{2}}\right)$

as desired.

Notes

’Prosthaphaeresis’ comes from the Greek: “prosthesi” = addition + “afairo” = subtraction.

The Prosthaphaeresis formula $\cos x\cos y=\frac{\cos (x+y)+\cos (x-y)}{2}$ was used by scientists to transform multiplication into addition. For example, to calculate the product $ab$, where (for $a$ and $b$ outside of this range, it is a simple matter to multiply or divide by a factor of 10 and divide or multiply this back in later), one would let $\cos x=a$ and $\cos y=b$. Using a table of cosines, one could then find an approximate value for $x$ and $y$, then find $x+y$ and $x-y$, and look up the cosines of the resulting two quantities (that is, $\cos (x+y)$ and $\cos (x-y)$). The average of these numbers is the desired product $ab$. This technique was used by Tycho Brahe to perform astronomical calculations.

Title	Prosthaphaeresis formulas
Canonical name	ProsthaphaeresisFormulas
Date of creation	2013-03-22 14:33:55
Last modified on	2013-03-22 14:33:55
Owner	mathfanatic (5028)
Last modified by	mathfanatic (5028)
Numerical id	7
Author	mathfanatic (5028)
Entry type	Proof
Classification	msc 26A09
Synonym	Simpson’s formulas