Question

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

Answer 1

Simplify the given expression as shown below

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

On the other hand,

`\begin{gathered} sin(y)=(1)/(2) \\ \end{gathered}`

Solving for y using the special triangle shown below

Thus,

`\begin{gathered} \Rightarrow y=30\degree\pm360\degree n=(\pi)/(6)\pm2\pi n \\ and \\ y=150\degree+360\degree n=(5\pi)/(6)+2\pi n \end{gathered}`

Then,

`\begin{gathered} \Rightarrow(\pi)/(6)x=y \\ \Rightarrow(\pi)/(6)x=(\pi)/(6)+2\pi n \\ \Rightarrow x=1+12n \\ and \\ (\pi)/(6)x=(5\pi)/(6)+2\pi n \\ \Rightarrow x=5+12n \end{gathered}`

The two sets of solutions are

`x=1+12n,5+12n`

Then, the four smallest positive solutions are

`\Rightarrow x=1,5,13,17`

The answers are 1,5,13,17