Q.Is $\sin^{-1}\!\left(\dfrac32\right)$ defined? Give a reason.

The domain of an inverse trigonometric function equals the range of the original trigonometric function on its chosen branch, and its range equals that branch itself. Since $\sin x$ and $\cos x$ never exceed $1$ in absolute value, $\sin^{-1}x$ and $\cos^{-1}x$ are defined only for $x\in[-1,1]$; since $\sec x$ and $\text{cosec}\,x$ never lie strictly between $-1$ and $1$ (from $\sec^2x\ge1$), $\sec^{-1}x$ and $\text{cosec}^{-1}x$ are defined only for $|x|\ge1$; $\tan^{-1}x$ and $\cot^{-1}x$, by contrast, are defined for every real $x$, since $\tan x$ and $\cot x$ take every real value on their principal branches.