Equation with sum-to-product formulae

Statement

Solve the equation using the sum-to-product formulae: $\sin x + \sin(3x) = 0$

Solution

We apply the sum-to-product formula $\ \sin p + \sin q = 2\sin\dfrac{p+q}{2}\cos\dfrac{p-q}{2}\$ with $p = 3x$ and $q = x$: $\sin(3x) + \sin x = 2\sin(2x)\cos x = 0.$ The product vanishes when one of the two factors vanishes:

• $\sin(2x) = 0 \Rightarrow 2x = k\pi \Rightarrow x = k\dfrac{\pi}{2}$;
• $\cos x = 0 \Rightarrow x = \dfrac{\pi}{2} + k\pi$, values already included in the previous family.

The solutions are therefore $\boxed{\,x = k\frac{\pi}{2},\quad k\in\mathbb{Z}.}$

Links

Topics: Trigonometric equations
Concepts: Elementary equation · Sum-to-product formulae
Functions: Cosine · Sine
Skills: Solving equations · Using formulae
Exercise type: Solving an equation