To determine the values of [tex]\(\theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] where the functions [tex]\(f(\theta) = 2 \cos^2 \theta\)[/tex] and [tex]\(g(\theta) = -1 - 4 \cos \theta - 2 \cos^2 \theta\)[/tex] intersect, we will follow these steps:
1. Set the two functions equal to each other:
[tex]\[
2 \cos^2 \theta = -1 - 4 \cos \theta - 2 \cos^2 \theta
\][/tex]
2. Combine like terms to simplify the equation:
[tex]\[
2 \cos^2 \theta + 2 \cos^2 \theta + 4 \cos \theta + 1 = 0
\][/tex]
Simplifying further:
[tex]\[
4 \cos^2 \theta + 4 \cos \theta + 1 = 0
\][/tex]
3. Divide the entire equation by 4 for simplicity:
[tex]\[
\cos^2 \theta + \cos \theta + \frac{1}{4} = 0
\][/tex]
4. Recognize that this is a quadratic equation in terms of [tex]\(\cos \theta\)[/tex]. Set [tex]\(y = \cos \theta\)[/tex], and rewrite the equation as:
[tex]\[
y^2 + y + \frac{1}{4} = 0
\][/tex]
5. Use the quadratic formula [tex]\(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = \frac{1}{4}\)[/tex]:
[tex]\[
y = \frac{-1 \pm \sqrt{1 - 1}}{2}
\][/tex]
6. Simplify the solution:
[tex]\[
y = \frac{-1 \pm 0}{2}
\][/tex]
Therefore:
[tex]\[
y = -\frac{1}{2}
\][/tex]
7. Recall that [tex]\(y = \cos \theta\)[/tex]. Hence:
[tex]\[
\cos \theta = -\frac{1}{2}
\][/tex]
8. Determine the values of [tex]\(\theta\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] where [tex]\(\cos \theta = -\frac{1}{2}\)[/tex]. These values are:
[tex]\[
\theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3}
\][/tex]
Therefore, the functions [tex]\(f(\theta)\)[/tex] and [tex]\(g(\theta)\)[/tex] intersect at:
[tex]\[
\theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3}
\][/tex]
So, the correct answer is:
[tex]\[
\theta = \frac{2\pi}{3}, \frac{4\pi}{3}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{\theta = \frac{2\pi}{3}, \frac{4\pi}{3}}
\][/tex]
