Question

question 10
solve \\( \cos ( x ) = 0.55 \\) 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
add work

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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:

\(A = 1.007\)
\(B = 5.276\)