Prosthaphaeresis formulas

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

sinA+sinB = 2sin(A+B2)cos(A-B2)
sinA-sinB = 2sin(A-B2)cos(A+B2)
cosA+cosB = 2cos(A+B2)cos(A-B2)
cosA-cosB = -2sin(A+B2)sin(A-B2)

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

sin(X+Y) = sinXcosY+cosXsinY
sin(X-Y) = sinXcosY-cosXsinY

Adding or subtracting the two equations yields

sin(X+Y)+sin(X-Y) = 2sinXcosY
sin(X+Y)-sin(X-Y) = 2sinYcosX

If we let X=A+B2 and Y=A-B2, then X+Y=2A2=A and X-Y=2B2=B, and the last two equations become

sinA+sinB = 2sin(A+B2)cos(A-B2)
sinA-sinB = 2sin(A-B2)cos(A+B2)

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) = cosXcosY-sinXsinY
cos(X-Y) = cosXcosY+sinXsinY

Adding or subtracting the two equations yields

cos(X+Y)+cos(X-Y) = 2cosXcosY
cos(X+Y)-cos(X-Y) = -2sinYsinX

Again, if we let X=A+B2 and Y=A-B2, then X+Y=2A2=A and X-Y=2B2=B, and the last two equations become

cosA+cosB = 2cos(A+B2)cos(A-B2)
cosA-cosB = -2sin(A-B2)sin(A+B2)

as desired.


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

The Prosthaphaeresis formula cosxcosy=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<a,b<1 (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 cosx=a and cosy=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 averageMathworldPlanetmath 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