Answer:
[tex]\begin{gathered} x_1=1 \\ x_2=13 \\ x_3=25 \\ x_4=37 \end{gathered}[/tex]
Step-by-step explanation:
Solving for x,
[tex]\begin{gathered} 8\sin(\frac{\pi}{6}x)=4 \\ \\ \rightarrow\sin(\frac{\pi}{6}x)=\frac{4}{8} \\ \\ \rightarrow\sin(\frac{\pi}{6}x)=\frac{1}{2} \\ \\ \rightarrow\frac{\pi}{6}x=\sin^{-1}(\frac{1}{2}) \\ \\ \rightarrow\frac{\pi}{6}x=\frac{\pi}{6} \end{gathered}[/tex]
Since sine has a periodicity of 2 pi,
[tex]\rightarrow\frac{\pi}{6}x=2\pi n+\frac{\pi}{6}[/tex]
Where n is any integer.
Furthermore,
[tex]\begin{gathered} \operatorname{\rightarrow}\frac{\pi}{6}x=2\pi n+\frac{\pi}{6} \\ \\ \rightarrow x=\frac{2\pi n}{\frac{\pi}{6}}+1 \\ \\ \Rightarrow x=12n+1 \end{gathered}[/tex]
Therefore, the four smallest positive solutions will correspond to:
[tex]n=0,1,2,3[/tex]
This way,
[tex]\begin{gathered} x_1=12(0)+1=1 \\ x_2=12(1)+1=13 \\ x_3=12(2)+1=25 \\ x_4=12(3)+1=37 \end{gathered}[/tex]
