To determine the values in the interval [tex]\([0, 2\pi)\)[/tex] at which the functions [tex]\( f(x) = \cos 2x + 2 \)[/tex] and [tex]\( g(x) = \sin x + 3 \)[/tex] intersect, we follow these steps:
1. Set up the equation [tex]\( f(x) = g(x) \)[/tex]:
[tex]\[
\cos 2x + 2 = \sin x + 3
\][/tex]
2. Simplify the equation:
[tex]\[
\cos 2x + 2 - \sin x - 3 = 0
\][/tex]
[tex]\[
\cos 2x - \sin x - 1 = 0
\][/tex]
3. Solve the equation for [tex]\( x \)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex]:
This requires finding the points where the two functions are equal in the specified interval.
4. List the points of intersection:
After working through the solution, the calculated points of intersection are:
[tex]\[
0, \pi, \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
5. Verify if these points fall within the given interval [tex]\([0, 2\pi)\)[/tex]:
- [tex]\( 0 \)[/tex] is in [tex]\([0, 2\pi)\)[/tex]
- [tex]\( \pi \)[/tex] is in [tex]\([0, 2\pi)\)[/tex]
- [tex]\( \frac{7\pi}{6} \)[/tex] (which is approximately 3.66519) is in [tex]\([0, 2\pi)\)[/tex]
- [tex]\( \frac{11\pi}{6} \)[/tex] (which is approximately 5.75959) is in [tex]\([0, 2\pi)\)[/tex]
Therefore, the values of [tex]\(x\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] at which the functions [tex]\( f(x) = \cos 2x + 2 \)[/tex] and [tex]\( g(x) = \sin x + 3 \)[/tex] intersect are:
[tex]\[
x = 0, \pi, \frac{7\pi}{6}, \frac{11\pi}{6}
\][/tex]
Hence, the correct multiple-choice option is:
[tex]\[
\boxed{x = 0, \pi, \frac{7\pi}{6}, \frac{11\pi}{6}}
\][/tex]
