Question

solve \\( \sin ( x ) = 0.46 \\) on \\( 0 \leq x < 2 \pi \\).
there are two solutions, \\( a \\) and \\( b \\), with \\( a < b \\).
\\( a = \\)
\\( b = \\)
give your answers accurate to 3 decimal places.
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Explanation:

Step1: Find the reference angle

We know that if \(\sin(x)=a\), then the reference angle \(x_0=\arcsin(a)\). Here \(a = 0.46\), so \(x_0=\arcsin(0.46)\).
Using a calculator, \(x_0=\arcsin(0.46)\approx0.470\) (in radians).

Step2: Find the first - quadrant solution (A)

Since \(\sin(x)\) is positive in the first and second quadrants. The solution in the first quadrant (\(0\leq x<\frac{\pi}{2}\)) is \(A = x_0\). So \(A=\arcsin(0.46)\approx0.470\).

Step3: Find the second - quadrant solution (B)

The solution in the second quadrant (\(\frac{\pi}{2}\leq x<\pi\)) is \(B=\pi - x_0\).
Substitute \(x_0 = 0.470\) into \(B=\pi - x_0\). Using \(\pi\approx3.142\), we get \(B = 3.142-0.470=2.672\).

Answer:

\(A = 0.470\)
\(B = 2.672\)