To solve the equation [tex]\(\cos(x + \pi) = \frac{1}{2}\)[/tex] over the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex], we can follow a series of steps to find the correct value(s) of [tex]\(x\)[/tex].
1. Understand the given equation:
We start with the equation:
[tex]\[\cos(x + \pi) = \frac{1}{2}\][/tex]
2. Use the properties of cosine to simplify the equation:
We know from trigonometric identities that:
[tex]\[\cos(x + \pi) = -\cos(x)\][/tex]
Therefore, we can rewrite the equation as:
[tex]\[-\cos(x) = \frac{1}{2}\][/tex]
Simplifying this gives:
[tex]\[\cos(x) = -\frac{1}{2}\][/tex]
3. Find the general solutions for [tex]\(\cos(x) = -\frac{1}{2}\)[/tex]:
We need to determine where the cosine function is equal to [tex]\(-\frac{1}{2}\)[/tex]. On the unit circle, the value of [tex]\(\cos(x) = -\frac{1}{2}\)[/tex] at [tex]\(x = \frac{2\pi}{3}\)[/tex] and [tex]\(x = \frac{4\pi}{3}\)[/tex] within the domain of one period, [tex]\(0 \leq x \leq 2\pi\)[/tex].
4. Check if the solutions lie within the given interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex]:
We need to see if our solutions fall within the specified interval:
- The specific interval we are looking at is [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex].
- The value [tex]\(x = \frac{2\pi}{3}\)[/tex] is within the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] since [tex]\(\frac{\pi}{2} < \frac{2\pi}{3} < \pi\)[/tex].
- The value [tex]\(x = \frac{4\pi}{3}\)[/tex] is not within the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] since [tex]\(\frac{4\pi}{3} > \pi\)[/tex].
5. Identify the valid solution:
From the above analysis, the only valid solution within the interval [tex]\(\left[\frac{\pi}{2}, \pi\right]\)[/tex] is:
[tex]\[x = \frac{2\pi}{3}\][/tex]
Thus, the value [tex]\(x = \frac{2\pi}{3}\)[/tex] satisfies the equation [tex]\(\cos(x + \pi) = \frac{1}{2}\)[/tex] over the interval [tex]\(\left[\frac{\pi}{2},\pi\right]\)[/tex]. The correct answer is:
[tex]\[ \boxed{\frac{2\pi}{3}} \][/tex]
