Question

solve \\( \cos ( x ) = 0.47 \\) on \\( 0 \leq x < 2 \pi \\).
there are two solutions, \\( a \\) and \\( b \\), with \\( a < b \\).
\\( a = \\)
\\( b = \\)

Explanation:

Step1: Find the principal solution

We know that if \(\cos(x) = a\), then \(x=\cos^{- 1}(a)+2k\pi\) or \(x = 2\pi-\cos^{-1}(a)+2k\pi\), \(k\in\mathbb{Z}\).
For \(a = 0.47\), the principal solution is \(x_1=\cos^{-1}(0.47)\).
Using a calculator, \(\cos^{-1}(0.47)\approx1.087\) (in radians).

Step2: Find the second - solution

The second solution in the interval \(0\leq x<2\pi\) is \(x_2 = 2\pi-\cos^{-1}(0.47)\).
\(x_2=2\pi - 1.087\approx6.283 - 1.087=5.196\)

Answer:

\(A\approx1.087\), \(B\approx5.196\)