Inverse trigonometric functions arcsin, arccos, arctan, arccot

Why “invert” $\sin$ , $\cos$ , $\tan$ , $\cot$

The functions $y=\arcsin x$ , $y=\arccos x$ , $y=\arctan x$ , $y=\operatorname{arccot}x$ in school are defined as inverses of restrictions of $\sin$ , $\cos$ , $\tan$ , $\cot$ to intervals of strict monotonicity where they are bijective onto their ranges. Then a relation like $\arcsin(\sin\alpha)=\alpha$ holds not always, but when $\alpha$ lies in the range of that particular inverse (the principal interval).

$\arcsin$ and $\arccos$ are defined only for $x\in[-1;1]$

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