PlanetMath: Prosthaphaeresis formulas

(more info)

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Prosthaphaeresis formulas

(Proof)

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 , where (for and 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 and , then find and , 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 . This technique was used by Tycho Brahe to perform astronomical calculations.