To solve the given equation [tex]\( 1 + \cos \theta = \frac{\sqrt{3} + 2}{2} \)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex], we will follow these steps:
1. Solve for [tex]\(\cos \theta\)[/tex]:
[tex]\[
1 + \cos \theta = \frac{\sqrt{3} + 2}{2}
\][/tex]
Subtract 1 from both sides to isolate [tex]\(\cos \theta\)[/tex]:
[tex]\[
\cos \theta = \frac{\sqrt{3} + 2}{2} - 1
\][/tex]
Simplify the right-hand side:
[tex]\[
\cos \theta = \frac{\sqrt{3} + 2}{2} - \frac{2}{2}
\][/tex]
[tex]\[
\cos \theta = \frac{\sqrt{3} + 2 - 2}{2}
\][/tex]
[tex]\[
\cos \theta = \frac{\sqrt{3}}{2}
\][/tex]
2. Determine [tex]\(\theta\)[/tex] values that satisfy [tex]\(\cos \theta = \frac{\sqrt{3}}{2}\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex]:
Recall that [tex]\(\cos \theta = \frac{\sqrt{3}}{2}\)[/tex] for specific angles in trigonometric functions. The angles for which cosine is [tex]\(\frac{\sqrt{3}}{2}\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{6} \quad \text{and} \quad \theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}
\][/tex]
3. Verify the possible solutions from the given options:
- Option A: [tex]\( \frac{\pi}{6}, \frac{11\pi}{6} \)[/tex]
- Option B: [tex]\(\frac{\pi}{6}, \frac{5\pi}{6}\)[/tex]
- Option C: [tex]\(\frac{\pi}{3}, \frac{5\pi}{3}\)[/tex]
- Option D: [tex]\(\frac{7\pi}{6}, \frac{11\pi}{6}\)[/tex]
We need to check which pair matches our determined values for [tex]\(\cos \theta = \frac{\sqrt{3}}{2}\)[/tex]:
The pair [tex]\( \frac{\pi}{6}, \frac{11\pi}{6} \)[/tex] are exactly the angles where [tex]\(\cos \theta\)[/tex] equals [tex]\(\frac{\sqrt{3}}{2}\)[/tex].
Thus, the correct answer is:
A. [tex]\( \frac{\pi}{6}, \frac{11\pi}{6} \)[/tex]
