solve \\( \\sin ( x ) = - 0.67 \\) on \\( 0 \\leq x < 2 \\pi \\)\nthere are two solutions, \\( a \\) and \\(…
solve \\( \\sin ( x ) = - 0.67 \\) on \\( 0 \\leq x < 2 \\pi \\)\nthere are two solutions, \\( a \\) and \\( b \\), with \\( a < b \\)\n\\( a = \\)\n\\( b = \\)\ngive your answers accurate to 3 decimal places
Answer
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)