To find the value of [tex]\(\cos^{-1}\left(-\frac{1}{2}\right)\)[/tex], we need to determine in which angle [tex]\( \theta \)[/tex] the cosine function equals [tex]\(-\frac{1}{2}\)[/tex].
The inverse cosine function, [tex]\(\cos^{-1}(x)\)[/tex], returns the principal value of the angle in the range [tex]\([0, \pi]\)[/tex] for which the cosine of the angle is [tex]\(x\)[/tex].
We are given:
[tex]\[
\cos \theta = -\frac{1}{2}
\][/tex]
To identify the angle [tex]\( \theta \)[/tex], recall where the cosine function takes the value [tex]\(-\frac{1}{2}\)[/tex]. The cosine of [tex]\(\frac{2\pi}{3}\)[/tex] radians (which is 120 degrees) is:
[tex]\[
\cos \left(\frac{2\pi}{3}\right) = -\frac{1}{2}
\][/tex]
Thus, the value of [tex]\(\theta\)[/tex] that satisfies [tex]\(\cos \theta = -\frac{1}{2}\)[/tex] in the range [tex]\([0, \pi]\)[/tex] is:
[tex]\[
\theta = \frac{2\pi}{3}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{2\pi}{3}}
\][/tex]
