To determine the value of [tex]\(\cos^{-1}(0)\)[/tex], we first need to understand that [tex]\(\cos^{-1}(x)\)[/tex] (also known as [tex]\(\arccos(x)\)[/tex]) represents the angle [tex]\(\theta\)[/tex] such that [tex]\(\cos(\theta) = x\)[/tex].
We are asked to find [tex]\(\cos^{-1}(0)\)[/tex]. This means we need to identify the angle [tex]\(\theta\)[/tex] for which
[tex]\[
\cos(\theta) = 0.
\][/tex]
The cosine function equals 0 at specific points within its range. The most common angles where the cosine function equals 0 are [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\frac{3\pi}{2}\)[/tex]. However, the principal value of [tex]\(\cos^{-1}(x)\)[/tex] is typically taken within the range [tex]\([0, \pi]\)[/tex].
In this range, the angle [tex]\(\theta\)[/tex] where [tex]\(\cos(\theta) = 0\)[/tex] is
[tex]\[
\theta = \frac{\pi}{2}.
\][/tex]
Thus, the value of [tex]\(\cos^{-1}(0)\)[/tex] is [tex]\(\frac{\pi}{2}\)[/tex] radians. Therefore, the correct answer is:
[tex]\[
\boxed{\frac{\pi}{2}}
\][/tex]
