Sum-to-product formulae: from sum to product

The sum-to-product (prosthaphaeresis) formulae carry out the inverse move to the Werner ones: they transform a sum or difference of sines and cosines into a product. They are obtained from the Werner formulae by a change of variable.

Property — Sum-to-product formulae: sum → product

Setting $p=\alpha+\beta$ and $q=\alpha-\beta$ (that is, $\alpha = (p+q)/2$ and $\beta=(p-q)/2$), from the Werner formulae one derives: $\begin{aligned} \sin p + \sin q &= 2\sin\!\tfrac{p+q}{2}\cos\!\tfrac{p-q}{2} \\ \sin p - \sin q &= 2\cos\!\tfrac{p+q}{2}\sin\!\tfrac{p-q}{2} \\ \cos p + \cos q &= 2\cos\!\tfrac{p+q}{2}\cos\!\tfrac{p-q}{2} \\ \cos p - \cos q &= -2\sin\!\tfrac{p+q}{2}\sin\!\tfrac{p-q}{2} \end{aligned}$

Since the result is a product, these formulae are the natural tool for factorising an expression and then applying the zero-product law.

Links

Topics: Prostaferesi werner
Concepts: Formule di prostaferesi · Formule di werner
Functions: Coseno · Seno
Methods: Formule prostaferesi
Skills: Scomporre · Usare formule