Question

assignment 7.5: solving trigonometric equations
score: 90/100 answered 10/12
× question 6
score on last try: 0 of 10 pts. see details for more.
at least one scored part is incorrect. jump to first changable incorrect part.

next question get a similar question you can retry this question below

solve \\( \cos ( x ) = 0.84 \\) 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 post to forum

Explanation:

Step1: Find the first solution

We know that if \(\cos(x)=a\), then \(x = \cos^{-1}(a)\) (principal value). Here \(a = 0.84\), so \(x=\cos^{-1}(0.84)\). Using a calculator, \(\cos^{-1}(0.84)\approx0.574\) (since \(0\leq x<2\pi\) and \(\cos(x)\) is positive in the first and fourth quadrants, the first - quadrant solution \(A\) is \(x=\cos^{-1}(0.84)\)).

Step2: Find the second solution

We use the identity \(\cos(x)=\cos(2\pi - x)\). The second - quadrant solution \(B\) is \(x = 2\pi-\cos^{-1}(0.84)\). Substitute \(\cos^{-1}(0.84)\approx0.574\) into \(2\pi - x\). So \(2\pi-0.574\approx2\times3.142 - 0.574=6.284 - 0.574 = 5.710\).

Answer:

\(A = 0.574\)
\(B = 5.710\)