Question

solve \\( \sin ( x ) = 0.8 \\) 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 \(x_{ref}=\sin^{- 1}(|a|)\). Here \(a = 0.8\), so \(x_{ref}=\sin^{-1}(0.8)\).
Using a calculator, \(x_{ref}\approx0.927\) (since \(\sin^{-1}(0.8)\approx0.927\) radians)

Step2: Find the first - quadrant solution (A)

In the interval \(0\leq x<2\pi\), for the sine function \(y = \sin(x)\), when \(x\) is in the first quadrant (\(0\leq x\leq\frac{\pi}{2}\)), \(x=x_{ref}\). So \(A=\sin^{-1}(0.8)\approx0.927\)

Step3: Find the second - quadrant solution (B)

We know that the sine function has the property \(\sin(x)=\sin(\pi - x)\). For \(x\) in the second quadrant (\(\frac{\pi}{2}\(B=\pi - 0.927\approx3.142-0.927 = 2.215\)

Answer:

\(A = 0.927\), \(B=2.215\)