Equation with sum-to-product formulae

Problem

Solve the equation $\sin(3x) = \sin x$ using the sum-to-product formulae.

Solution

We move everything to the left-hand side and apply the sum-to-product formula $\sin p - \sin q = 2\cos\!\tfrac{p+q}{2}\sin\!\tfrac{p-q}{2}$ with $p=3x$ and $q=x$: $\sin(3x) - \sin x = 2\cos(2x)\sin x = 0.$ By the zero-product law the equation splits into two: $\cos(2x) = 0 \quad\Longrightarrow\quad 2x = \tfrac{\pi}{2}+k\pi \quad\Longrightarrow\quad x = \tfrac{\pi}{4}+k\tfrac{\pi}{2};$ $\sin x = 0 \quad\Longrightarrow\quad x = k\pi.$ The solutions are therefore $\boxed{x = \tfrac{\pi}{4}+k\tfrac{\pi}{2}\ \lor\ x = k\pi}$, with $k\in\mathbb{Z}$.

Links

Topics: Prostaferesi werner
Concepts: Formule di prostaferesi · Legge di annullamento del prodotto
Functions: Coseno · Seno
Methods: Formule prostaferesi
Skills: Risolvere equazioni · Scomporre
Exercise type: Risoluzione equazione