Simplify the given expression as shown below
[tex]\begin{gathered} 8sin(\frac{\pi}{6}x)=4 \\ \Rightarrow sin(\frac{\pi}{6}x)=\frac{4}{8}=\frac{1}{2} \\ \Rightarrow sin(\frac{\pi}{6}x)=\frac{1}{2} \end{gathered}[/tex]
On the other hand,
[tex]\begin{gathered} sin(y)=\frac{1}{2} \\ \end{gathered}[/tex]
Solving for y using the special triangle shown below
Thus,
[tex]\begin{gathered} \Rightarrow y=30\degree\pm360\degree n=\frac{\pi}{6}\pm2\pi n \\ and \\ y=150\degree+360\degree n=\frac{5\pi}{6}+2\pi n \end{gathered}[/tex]
Then,
[tex]\begin{gathered} \Rightarrow\frac{\pi}{6}x=y \\ \Rightarrow\frac{\pi}{6}x=\frac{\pi}{6}+2\pi n \\ \Rightarrow x=1+12n \\ and \\ \frac{\pi}{6}x=\frac{5\pi}{6}+2\pi n \\ \Rightarrow x=5+12n \end{gathered}[/tex]
The two sets of solutions are
[tex]x=1+12n,5+12n[/tex]
Then, the four smallest positive solutions are
[tex]\Rightarrow x=1,5,13,17[/tex]
The answers are 1,5,13,17
