## Task $8 / 72$

Determine all real numbers $a$ for which the equation

$$
\sin x + \sin (x + a) + \sin (x + 2a) = 0
$$

holds for any real $x$.
By applying the addition theorems, the given equation is equivalently transformed into

$$
\sin (x+a) \cdot(1+2 \cos a)=0
$$

Since $\sin (x+a)=0$ does not hold for arbitrary real $x$, $1+2 \cos a=0$ must be true for the equation to be satisfied. This immediately yields

$$
\cos a=-0.5 \rightarrow a_{1}=2\left(\frac{1}{3}+k\right) \pi ; a_{2}=2\left(\frac{2}{3}+k\right) \pi
$$

(where $k$ is any integer). The verification confirms the correctness.

Remark: This equation finds its practical application in three-phase alternating current.