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Prosthaphaeresis Formulas


The Prosthaphaeresis formulas, also known as Simpson's formulas, are trigonometry formulas that convert a product of functions into a sum or difference. They are given by

sinalpha+sinbeta=2sin[1/2(alpha+beta)]cos[1/2(alpha-beta)]
(1)
sinalpha-sinbeta=2cos[1/2(alpha+beta)]sin[1/2(alpha-beta)]
(2)
cosalpha+cosbeta=2cos[1/2(alpha+beta)]cos[1/2(alpha-beta)]
(3)
cosalpha-cosbeta=-2sin[1/2(alpha+beta)]sin[1/2(alpha-beta)].
(4)

This form of trigonometric functions can be obtained in the Wolfram Language using the command TrigFactor[expr].

TrigSumProduct

These can be derived using the above figure (Kung 1996). From the figure, define

theta=1/2(alpha-beta)
(5)
gamma=1/2(alpha+beta).
(6)

Then we have the identity

s=1/2(sinalpha+sinbeta)
(7)
=cos[1/2(alpha-beta)]sin[1/2(alpha+beta)]
(8)
t=1/2(cosalpha+cosbeta)
(9)
=cos[1/2(alpha-beta)]cos[1/2(alpha+beta)].
(10)
TrigDiffProduct

Trigonometric product formulas for the difference of the cosines and sines of two angles can be derived using the similar figure illustrated above (Kung 1996). With theta and gamma as previously defined, the above figure gives

u=cosbeta-cosalpha
(11)
=2sin[1/2(alpha-beta)]sin[1/2(alpha+beta)]
(12)
v=sinalpha-sinbeta
(13)
=2sin[1/2(alpha-beta)]cos[1/2(alpha+beta)].
(14)

See also

Harmonic Addition Theorem, Trigonometric Addition Formulas, Werner Formulas

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References

Kung, S. H. "Proof without Words: The Difference-Product Identities" and "Proof without Words: The Sum-Product Identities." Math. Mag. 69, 269, 1996.

Referenced on Wolfram|Alpha

Prosthaphaeresis Formulas

Cite this as:

Weisstein, Eric W. "Prosthaphaeresis Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProsthaphaeresisFormulas.html

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