Question

solve ( 6 sin left( \frac { pi } { 5 } x
ight) = 4 ) for the four smallest positive solutions.

( x = )

give your answers accurate to at least two decimal places, as a list separated by commas.

question help: video

Explanation:

Step1: Isolate the sine function

Divide both sides of the equation \(6\sin(\frac{\pi}{5}x)=4\) by \(6\) to get \(\sin(\frac{\pi}{5}x)=\frac{4}{6}=\frac{2}{3}\).

Step2: Use the inverse - sine function

We know that if \(\sin\theta = a\), then \(\theta=\sin^{- 1}(a)+2k\pi\) or \(\theta=\pi-\sin^{-1}(a)+2k\pi\), where \(k\in\mathbb{Z}\).
For \(\theta = \frac{\pi}{5}x\) and \(a=\frac{2}{3}\), we have \(\frac{\pi}{5}x=\sin^{-1}(\frac{2}{3}) + 2k\pi\) or \(\frac{\pi}{5}x=\pi-\sin^{-1}(\frac{2}{3})+2k\pi\).
First, find \(\sin^{-1}(\frac{2}{3})\approx0.73\) (in radians).

Case 1: \(\frac{\pi}{5}x=\sin^{-1}(\frac{2}{3})+2k\pi\)

Solve for \(x\): \(x=\frac{5}{\pi}(\sin^{-1}(\frac{2}{3}) + 2k\pi)\).
When \(k = 0\), \(x=\frac{5}{\pi}\sin^{-1}(\frac{2}{3})\approx\frac{5\times0.73}{3.14}\approx1.17\).
When \(k = 1\), \(x=\frac{5}{\pi}(\sin^{-1}(\frac{2}{3})+2\pi)=\frac{5}{\pi}\sin^{-1}(\frac{2}{3}) + 10\approx1.17 + 10=11.17\).

Case 2: \(\frac{\pi}{5}x=\pi-\sin^{-1}(\frac{2}{3})+2k\pi\)

Solve for \(x\): \(x=\frac{5}{\pi}(\pi-\sin^{-1}(\frac{2}{3})+2k\pi)=5-\frac{5}{\pi}\sin^{-1}(\frac{2}{3})+10k\).
When \(k = 0\), \(x = 5-\frac{5}{\pi}\sin^{-1}(\frac{2}{3})\approx5 - 1.17=3.83\).
When \(k = 1\), \(x=5-\frac{5}{\pi}\sin^{-1}(\frac{2}{3})+10\approx3.83+10 = 13.83\).

Answer:

\(1.17,3.83,11.17,13.83\)