If \(y=\sin^{-1}x\), which relation between (x) and (y) is correct?

Correct answer: \(x=\sin y,\; y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)

\(y=\sin^{-1}x\) means that \(y\) is the principal-value angle whose sine is \(x\); hence \(x=\sin y\). The principal range of \(\sin^{-1}x\) is \(\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\), so option A is correct. Option D reverses the relation between \(\sin\) and \(\sin^{-1}\). Exam tip: read \(\sin^{-1}x\) as the inverse sine function, not as \(1/\sin x\).