Question

give the smallest two solutions of \\( \sin ( 4 \theta ) = 0.2694 \\) on \\( 0,2 \pi ) \\).
separate the two solutions with a comma.

Explanation:

Step1: Solve for \(4\theta\)

We know that if \(\sin(4\theta)=0.2694\), then \(4\theta=\sin^{- 1}(0.2694)+2k\pi\) or \(4\theta=\pi-\sin^{-1}(0.2694)+2k\pi\), where \(k\in\mathbb{Z}\).
Using a calculator, \(\sin^{-1}(0.2694)\approx0.272\) (in radians).
So \(4\theta\approx0.272 + 2k\pi\) or \(4\theta\approx\pi - 0.272+2k\pi\approx2.87 + 2k\pi\).

Step2: Solve for \(\theta\)

For \(4\theta\approx0.272+2k\pi\), when \(k = 0\), \(\theta=\frac{0.272}{4}=0.068\).
For \(4\theta\approx2.87+2k\pi\), when \(k = 0\), \(\theta=\frac{2.87}{4}=0.7175\).

Answer:

\(0.068,0.7175\)