Question

solve \\( \sin ( x ) = - 0.67 \\) 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

Explanation:

Step1: Find the reference angle

Since \(\sin(x)= - 0.67\), first find the reference angle \(x_{ref}\) using the inverse sine function. \(x_{ref}=\sin^{-1}(0.67)\). Using a calculator, \(x_{ref}\approx0.735\) radians.

Step2: Determine the solutions in the given interval

The sine function is negative in the third and fourth quadrants.

• For the third - quadrant solution (\(A\)): \(x=\pi + x_{ref}\). Substitute \(x_{ref}\approx0.735\), then \(x=\pi+ 0.735\approx3.142 + 0.735 = 3.877\).
• For the fourth - quadrant solution (\(B\)): \(x = 2\pi-x_{ref}\). Substitute \(x_{ref}\approx0.735\), then \(x=2\pi - 0.735\approx6.283-0.735 = 5.548\).

Answer:

\(A = 3.877\), \(B = 5.548\)