1. **Problem statement:** Find the solutions to the equation $\sin(x) = 0.4$ in the interval $[0^\circ; 360^\circ]$. 2. **Formula and rules:** The sine function is periodic with period $360^\circ$. For $\sin(x) = a$, where $a$ is between $-1$ and $1$, the general solutions in degrees are: $$ x = \arcsin(a) \quad \text{and} \quad x = 180^\circ - \arcsin(a) $$ These two solutions lie within one period $[0^\circ, 180^\circ]$. Additional solutions can be found by adding multiples of $360^\circ$. 3. **Calculate the principal value:** $$ \arcsin(0.4) = \theta $$ where $\theta$ is the angle whose sine is 0.4. 4. **Find the two solutions in $[0^\circ, 360^\circ]$:** $$ x_1 = \theta $$ $$ x_2 = 180^\circ - \theta $$ 5. **Explain:** These two angles correspond to the points on the sine curve where the value is 0.4, one in the first quadrant and one in the second quadrant. 6. **Final answer:** The solutions to $\sin(x) = 0.4$ in $[0^\circ; 360^\circ]$ are $x = \theta$ and $x = 180^\circ - \theta$ where $\theta = \arcsin(0.4)$. Note: The exact numeric values are not calculated here as per the instruction to omit solutions and drawings.