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When trying to achieve precise positioning of labels, I sometimes see the label position not respond as I would expect to small changes in the position angle. Can someone explain what is going on?

See for instance the MWE. On the second node, the label does not seem to have shifted right even though the angle is 88 instead of 90.

And more flagrantly, on the third label, the label is shifted rightward by a lot even though the angle is just 87.

\documentclass{standalone}
\usepackage{tikz}
\tikzset{
dot/.style = {circle, fill, minimum size=4pt,
           inner sep=0pt, outer sep=0pt},
}
\begin{document}
\begin{tikzpicture}
\node[dot,label={90:A}] (first) at (0,0){};
\node[dot,label={88:A}] (second) at (1,0){};
\node[dot,label={87:A}] (third) at (2,0){};
\end{tikzpicture}
\end{document}

I suspect this might have to do with a change in the label anchor used to compute the angle, but it is definitely not what I would have expected as a behaviour.

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That's strange but expected. In the manual, page 247:

enter image description here

enter image description here

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