Question

solve \\( \sin ( x ) = 0.66 \\) 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 \(a = 0.66\), so \(x_{ref}=\arcsin(0.66)\).
Using a calculator, \(x_{ref}\approx0.718\) (in 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 (\(0\leq x\leq\frac{\pi}{2}\)), \(x = \arcsin(0.66)\). So \(A=\arcsin(0.66)\approx0.718\).

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}\(B=\pi - 0.718\approx3.142-0.718 = 2.424\)

Answer:

\(A = 0.718\)
\(B=2.424\)