# Prosthaphaeresis formulas

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

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

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

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

Adding or subtracting the two equations yields

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

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

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

as desired.

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

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

Adding or subtracting the two equations yields

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

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

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

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 $0 (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 ProsthaphaeresisFormulas 2013-03-22 14:33:55 2013-03-22 14:33:55 mathfanatic (5028) mathfanatic (5028) 7 mathfanatic (5028) Proof msc 26A09 Simpson’s formulas