<s>[INST] Solving $\sin 3\theta=1/2$ on the interval $[0,2\pi]$. I don't understand where some solutions came from. I am learning precalculus, and I understand how to obtain first two solutions, but I don't understand where did last four solutions came from:

All values of $\theta$ in the interval $[0,2\pi]$ that satisfy $\sin 3\theta=1/2$ are
  $$\theta = \frac{\pi}{18}, \frac{5\pi}{18}, \frac{13\pi}{18}, \frac{17\pi}{18}, \frac{25\pi}{18}, \frac{29\pi}{18}$$

 [/INST] Consider the diagram below.

When does $\sin\theta = \sin\varphi$?
The sine of a directed angle is the $y$-coordinate of the point where its terminal side intersects the unit circle.  Clearly, one solution is when $\theta = \varphi$.  By symmetry, $\sin\theta = \sin(\pi - \theta)$, so $\varphi = \pi - \theta$ is another solution.  Moreover, any angle coterminal with $\theta$ or $\pi - \theta$ will also have the same sine.  Since coterminal angles differ by integer multiples of $2\pi$, $\sin\theta = \sin\varphi$ if 
$$\varphi = \theta + 2k\pi, k \in \mathbb{Z}$$
or 
$$\varphi = \pi - \theta + 2m\pi, m \in \mathbb{Z}$$

Solve the equation $\sin(3\theta) = \dfrac{1}{2}$ in the interval $[0, 2\pi]$.

A particular solution of the equation 
$$\sin\alpha = \frac{1}{2}$$
is 
$$\alpha = \arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$$
Substituting $3\theta$ for $\alpha$ yields
$$\sin(3\theta) = \sin\left(\frac{\pi}{6}\right)$$
Thus, we have the general solution
\begin{align*}
3\theta & = \frac{\pi}{6} + 2k\pi, k \in \mathbb{Z} & 3\theta & = \pi - \frac{\pi}{6} + 2m\pi, m \in \mathbb{Z}\\
\theta & = \frac{\pi}{18} + \frac{2k\pi}{3}, k \in \mathbb{Z} & 3\theta & = \frac{5\pi}{6} + 2m\pi, m \in \mathbb{Z}\\
& & \theta & = \frac{5\pi}{18} + \frac{2m\pi}{3}, m \in \mathbb{Z}
\end{align*}
Since we seek solutions in the interval $[0, 2\pi]$, $k = 0, 1, 2$ and $m = 0, 1, 2$, which yields
\begin{align*}
\theta & = \frac{\pi}{18}, \frac{13\pi}{18}, \frac{25\pi}{18} & \theta & = \frac{5\pi}{18}, \frac{17\pi}{18}, \frac{29\pi}{18}
\end{align*}
</s>