Question

Solve 8 sin ( pi 6 x )=4 1 for the four smallest positive solutions

Answer 1

`\begin{gathered} x_1=1 \\ x_2=13 \\ x_3=25 \\ x_4=37 \end{gathered}`

Explanation:

Solving for x,

`\begin{gathered} 8\sin((\pi)/(6)x)=4 \\ \\ \rightarrow\sin((\pi)/(6)x)=(4)/(8) \\ \\ \rightarrow\sin((\pi)/(6)x)=(1)/(2) \\ \\ \rightarrow(\pi)/(6)x=\sin^(-1)((1)/(2)) \\ \\ \rightarrow(\pi)/(6)x=(\pi)/(6) \end{gathered}`

Since sine has a periodicity of 2 pi,

`\rightarrow(\pi)/(6)x=2\pi n+(\pi)/(6)`

Where n is any integer.

Furthermore,

Therefore, the four smallest positive solutions will correspond to:

`n=0,1,2,3`

This way,

`\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}`