If we just restrict ourselves to [0, 2pi], the sine will equal .5 at pi/6 and 5p/6 rads
So we have.... (pi/4(x-6))= pi/6 → pi/4 *x - 3pi/2 = pi/6 → pi/4 * x = [5pi/3]/ → x = [5pi/3][4/pi]
→ 20/3
Also .... (pi/4(x-6)) = 5pi/6 → pi/4 *x - 3pi/2 = 5pi/6 → pi/4 * x = [7 pi/3] → x = [7pi/3][4/pi] → 28/3
If we just restrict ourselves to [0, 2pi], the sine will equal .5 at pi/6 and 5p/6 rads
So we have.... (pi/4(x-6))= pi/6 → pi/4 *x - 3pi/2 = pi/6 → pi/4 * x = [5pi/3]/ → x = [5pi/3][4/pi]
→ 20/3
Also .... (pi/4(x-6)) = 5pi/6 → pi/4 *x - 3pi/2 = 5pi/6 → pi/4 * x = [7 pi/3] → x = [7pi/3][4/pi] → 28/3
$$\begin{array}{rlll}
sin (\frac{\pi(x-6)}{4})&=&0.5\\\\
\frac{\pi(x-6)}{4}&=&n\pi +(-1)^n\times \frac{\pi}{6}\qquad & n\in Z\\\\
x-6&=&4n +(-1)^n\times \frac{4}{6}\qquad & n\in Z\\\\
x&=&4n+6 +(-1)^n\times \frac{2}{3}\qquad & n\in Z\\\\
x&=&4n+6 + \frac{(-1)^n\times 2}{3}\qquad & n\in Z\\\\
\end{array}$$
I guess that is the domain since that is all the values that x can be :/