Every Solution of sin 3x

For every solution of $\sin\left(2x\right) = \frac{1}{2}$, write the families for 2x first: $2x = 30^{\circ}+360^{\circ}\cdot k$ or $2x = 150^{\circ}+360^{\circ}\cdot k$. Then divide everything by 2, the period too:

$$x = 15^{\circ}+180^{\circ}\cdot k$$

$$x = 75^{\circ}+180^{\circ}\cdot k$$

The most common slip is halving the 30° and keeping 360°k. That family misses 195° and every other answer. A multiple angle shrinks the period: sin 2x repeats every 180°, sin 3x every 120°.