Question

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

Answer 1

Simplify the given expression as shown below

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

On the other hand,

$\begin{gathered} sin(y)=\frac{1}{2} \\ \end{gathered}$

Solving for y using the special triangle shown below

Thus,

$\begin{gathered} \Rightarrow y=30\degree\pm360\degree n=\frac{\pi}{6}\pm2\pi n \\ and \\ y=150\degree+360\degree n=\frac{5\pi}{6}+2\pi n \end{gathered}$

Then,

$\begin{gathered} \Rightarrow\frac{\pi}{6}x=y \\ \Rightarrow\frac{\pi}{6}x=\frac{\pi}{6}+2\pi n \\ \Rightarrow x=1+12n \\ and \\ \frac{\pi}{6}x=\frac{5\pi}{6}+2\pi n \\ \Rightarrow x=5+12n \end{gathered}$

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