Factorising a sum into a product

An example shows how the sum-to-product formulae turn a sum into a product and open the way to solving an equation.

Example — Factorising

Factorise $\sin(3x) + \sin x$ into a product.

With $p=3x$ and $q=x$: $\sin(3x) + \sin x = 2\sin\!\left(\tfrac{3x+x}{2}\right)\cos\!\left(\tfrac{3x-x}{2}\right) = 2\sin(2x)\cos x.$ This factorisation is extremely useful for solving the equation $\sin(3x)+\sin x = 0$: applying the sum-to-product formula it becomes $2\sin(2x)\cos x = 0$, which by the zero-product law splits into the two elementary equations $\sin(2x)=0$ and $\cos x=0$.

Links

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