To solve the equation [tex]\( 1 + \cos \theta = \frac{1}{2} \)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Isolate [tex]\(\cos \theta\)[/tex]:
[tex]\[
1 + \cos \theta = \frac{1}{2}
\][/tex]
Subtract 1 from both sides of the equation:
[tex]\[
\cos \theta = \frac{1}{2} - 1
\][/tex]
Simplify the right-hand side:
[tex]\[
\cos \theta = -\frac{1}{2}
\][/tex]
2. Find the general solutions to [tex]\(\cos \theta = -\frac{1}{2}\)[/tex]:
- The cosine function achieves the value [tex]\(-\frac{1}{2}\)[/tex] at specific angles within one full rotation [tex]\([0, 2\pi)\)[/tex].
- Cosine of [tex]\(\theta\)[/tex] is [tex]\(-\frac{1}{2}\)[/tex] at:
[tex]\[
\theta = \frac{2\pi}{3} \quad \text{(in the second quadrant)},
\][/tex]
[tex]\[
\theta = \frac{4\pi}{3} \quad \text{(in the third quadrant)}.
\][/tex]
3. Verify the angles are within the correct interval:
Both angles [tex]\( \frac{2\pi}{3} \)[/tex] and [tex]\( \frac{4\pi}{3} \)[/tex] are within the interval [tex]\([0, 2\pi)\)[/tex]. Therefore, they are valid solutions.
4. Summarize the solutions:
The solutions for the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{2\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{3}
\][/tex]
Thus, the correct answer is:
D. [tex]\(\frac{2\pi}{3}, \frac{4\pi}{3}\)[/tex]
