question 10\nsolve \\( \\cos ( x ) = 0.55 \\) on \\( 0 \\leq x < 2 \\pi \\)\nthere are two solutions, a and…

question 10\nsolve \\( \\cos ( x ) = 0.55 \\) 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\nadd work\n> next question

question 10\nsolve \\( \\cos ( x ) = 0.55 \\) 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\nadd work\n> next question

Answer

Explanation:

Step1: Find the first solution

We know that if (\cos(x)=a), then (x = \cos^{-1}(a)) (principal value). Given (a = 0.55), so (x=\cos^{-1}(0.55)). Using a calculator, (\cos^{-1}(0.55)\approx1.007) radians. This is the solution in the first - quadrant ((A)).

Step2: Find the second solution

We use the identity (\cos(x)=\cos(2\pi - x)). The second solution (B) in the interval (0\leq x<2\pi) is (B = 2\pi-\cos^{-1}(0.55)). Since (\cos^{-1}(0.55)\approx1.007) and (2\pi\approx6.283), then (B=6.283 - 1.007=5.276)

Answer:

(A = 1.007) (B = 5.276)