Question

solve \\( \cos ( x ) = 0.65 \\) 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.
question help: video

Explanation:

Step1: Find the reference angle

The reference angle \(x_0=\cos^{-1}(0.65)\). Using a calculator, \(x_0\approx0.863\) radians.

Step2: Find the first solution \(A\)

Since \(\cos(x)\) is positive in the first quadrant, \(A = x_0\approx0.863\) radians.

Step3: Find the second solution \(B\)

Since \(\cos(x)=\cos(2\pi - x)\), and \(\cos(x)\) is positive in the fourth quadrant. So \(B=2\pi - x_0\). Substitute \(x_0 = 0.863\), then \(B=2\pi- 0.863\approx5.420\) radians.

Answer:

\(A = 0.863\)
\(B = 5.420\)