Symbolic Notation for $\theta "=" \arcsin(-.5)$? I'm teaching PreCalculus and the following issue has always bugged me.
Problem: Solve $\sin\theta = -.5$ for $0 \le \theta \le 2\pi$.
Solution:
\begin{align*}
\sin\theta &= -.5\\
\theta &= \arcsin(-.5) = -\frac\pi6
\end{align*}
But to get this into our desired domain, our solutions are $\boxed{\theta = \frac{7\pi}6 \text{ or }\frac{11\pi}6}$.
So my objection is the line $\theta = \arcsin(-.5)$, because that's really not true. $\theta$ can be a whole lot of things! So does there exist some symbol or notation that expresses this? Something a-la "If $x^2=9$, then $x = \pm 3$." Like
$$\theta \stackrel{\text{is related to}}{\sim} \arcsin(-.5) = -\frac\pi6 ?$$

The way I always explained it to my students was "First, you find a solution, then you find the solution."
Let $\theta_0 = \arcsin(-\frac 12)$.

The range of $\arcsin$ is $-\frac{\pi}{2} \le \theta_0 \le \frac{\pi}{2}$. Since, on the unit circle, $\sin \theta = y=-\frac 12$, we see quickly that 
$\theta_0 = -\frac{\pi}{6}$. The two red dots indicate the two points on the unit circle for which $\sin \theta = -\frac 12$. One way to express this is
$\theta \in \{ -\frac{\pi}{6} + 2n\pi : n \in \mathbb Z\} \cup
            \{ \frac{7\pi}{6} + 2n\pi : n \in \mathbb Z\}$
To achieve $0 \le \theta \le 2\pi$ we find that
$\theta \in \{\frac{7\pi}{6}, \frac{11\pi}{6} \}$
where $ \frac{11\pi}{6} = -\frac{\pi}{6} + 2\pi$
