To find the equivalent value of [tex]\(\sin^{-1}\left(\cos\left(\frac{\pi}{2}\right)\right)\)[/tex], let's follow these steps carefully:
1. Evaluate the cosine function:
- We need to determine [tex]\(\cos\left(\frac{\pi}{2}\right)\)[/tex].
- The cosine of [tex]\(\frac{\pi}{2}\)[/tex] radians (or 90 degrees) is 0.
[tex]\[
\cos\left(\frac{\pi}{2}\right) = 0
\][/tex]
2. Apply the inverse sine function:
- Now we need to find [tex]\(\sin^{-1}(0)\)[/tex].
- The inverse sine function returns the angle whose sine is the given value. We need to find an angle [tex]\( \theta \)[/tex] such that [tex]\(\sin(\theta) = 0\)[/tex].
- Within the range of the inverse sine function [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex], the angle that satisfies [tex]\(\sin(\theta) = 0\)[/tex] is [tex]\( \theta = 0 \)[/tex].
3. Conclusion:
- Therefore, [tex]\(\sin^{-1}(0) = 0\)[/tex], and consequently:
[tex]\[
\sin^{-1}\left(\cos\left(\frac{\pi}{2}\right)\right) = \sin^{-1}(0) = 0
\][/tex]
Hence, the equivalent value of [tex]\(\sin^{-1}\left(\cos\left(\frac{\pi}{2}\right)\right)\)[/tex] in radians is [tex]\(0\)[/tex].
