To determine which value is equivalent to [tex]\(\sin^{-1}(0)\)[/tex], we need to understand what [tex]\(\sin^{-1}(x)\)[/tex] represents. The notation [tex]\(\sin^{-1}(x)\)[/tex] represents the inverse sine function, also known as arcsine, which returns the angle whose sine is [tex]\(x\)[/tex].
In this case, [tex]\(\sin^{-1}(0)\)[/tex] means we are asking for the angle [tex]\(\theta\)[/tex] such that [tex]\(\sin(\theta) = 0\)[/tex].
Let's consider the possible candidates:
1. [tex]\(0\)[/tex]: Check if [tex]\(\sin(0) = 0\)[/tex]:
[tex]\[
\sin(0) = 0
\][/tex]
This is true, so 0 is a candidate.
2. [tex]\(\frac{\pi}{2}\)[/tex]: Check if [tex]\(\sin\left(\frac{\pi}{2}\right) = 0\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{2}\right) = 1
\][/tex]
This is not true, so [tex]\(\frac{\pi}{2}\)[/tex] is not [tex]\(\sin^{-1}(0)\)[/tex].
3. [tex]\(\frac{3\pi}{2}\)[/tex]: Check if [tex]\(\sin\left(\frac{3\pi}{2}\right) = 0\)[/tex]:
[tex]\[
\sin\left(\frac{3\pi}{2}\right) = -1
\][/tex]
This is not true, so [tex]\(\frac{3\pi}{2}\)[/tex] is not [tex]\(\sin^{-1}(0)\)[/tex].
4. [tex]\(\frac{5\pi}{2}\)[/tex]: Check if [tex]\(\sin\left(\frac{5\pi}{2}\right) = 0\)[/tex]:
[tex]\[
\sin\left(\frac{5\pi}{2}\right) = 1
\][/tex]
This is not true, so [tex]\(\frac{5\pi}{2}\)[/tex] is not [tex]\(\sin^{-1}(0)\)[/tex].
Since the only angle that satisfies [tex]\(\sin(\theta) = 0\)[/tex] among the given choices is [tex]\(0\)[/tex], we conclude that:
[tex]\[
\sin^{-1}(0) = 0 \text{ radians}
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{0}
\][/tex]
