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.
question help: video

Explanation:

Step1: Find the reference angle

We know that if \(\sin(x) = a\), then the reference angle \(x_{ref}=\arcsin(a)\). Here \(a = 0.46\), so \(x_{ref}=\arcsin(0.46)\). Using a calculator, \(x_{ref}\approx0.479\) (in radians).

Step2: Find the first - quadrant solution (A)

Since the sine function is positive in the first and second quadrants. In the first quadrant (\(0\leq x<\frac{\pi}{2}\)), \(A = x_{ref}\). So \(A=\arcsin(0.46)\approx0.479\).

Step3: Find the second - quadrant solution (B)

In the second quadrant (\(\frac{\pi}{2}\leq x<\pi\)), the formula for the solution of \(\sin(x)=\sin(\alpha)\) is \(x=\pi - \alpha\). Here \(\alpha=\arcsin(0.46)\), so \(B=\pi-\arcsin(0.46)\).

$$B=\pi - 0.479\approx3.142-0.479 = 2.663$$

Answer:

\(A = 0.479\)
\(B = 2.663\)