Answer
The corresponding values of $x$ are $$\{\frac{\pi}{8}, \frac{5\pi}{6}, \frac{5\pi}{4}\}$$
Work Step by Step
1) The question asks for solutions over the interval $[0,2\pi)$, meaning that
$$x\in[0,2\pi)$$
or $$0\le x\lt2\pi$$
2) Now from also the question, your work leads to $$\frac{1}{2}x=\frac{\pi}{16}, \frac{5\pi}{12}, \frac{5\pi}{8}$$
To find corresponding values of $x$ from $\frac{1}{2}x$, we multiply both sides by $2$
$$x=\frac{\pi}{8}, \frac{5\pi}{6}, \frac{5\pi}{4}$$
3) Finally, it is crucial to compare the values of $x$ in 2) with the range found in 1).
We see that all three values of $x=\frac{\pi}{8}, \frac{5\pi}{6}, \frac{5\pi}{4}$ fit in the range of $x\in[0,2\pi)$. None would be eliminated as a result.
The corresponding values of $x$ are $$\{\frac{\pi}{8}, \frac{5\pi}{6}, \frac{5\pi}{4}\}$$
