Obviously (;-) @egreg’s answer is great and goes right to the point, but it does have an infinitesimal drawback: a pervert who tried $\overline{\sin x}$
would get a surprise. Moreover, one could argue that the placement of the radical sign in $\sqrt{\sin x}$
is suboptimal.
The positioning of superscripts in math formulas is a very low-level feature of TeX, which is described, along with many other “intimate” details of math typesetting, in Appendix G of The TeXbook, to which you are referred for an explanation of the following solution.
\documentclass[a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
% egreg's recipe:
\makeatletter
\DeclareRobustCommand{\sinname}{%
\smash{\operator@font sin}\vphantom{s}%
}
% ... but let us use a different name for the operator:
\newcommand{\ssin}{\qopname\relax o{\sinname}}
\makeatother
% Our recipe:
\setbox0 = \hbox{$$} % load math fonts
\fontdimen18\scriptfont2 = 3.78970pt % turns out to be just enough
\begin{document}
The downside in egreg's answer:
\[\sqrt{\sin x}+\sqrt{\ssin x} + \overline{\sin x}+\overline{\ssin x}\]
That's not fine.
But, with our correction, the exponent of the ``usual'' \verb|\sin| will not be
rised that much:
\[ \frac{1}{\cos^{2}x} + \frac{1}{\sin^{2}x} + \frac{1}{x^{2}} \]
For an explanation, see \textsl{The \TeX book}, Appendix~G, Rule~18a; the
relevant passage is ``set \( u\gets h-q \)\,\ldots\ where $q$\,\ldots\ [is] the
[value] of~$\sigma_{18}$\,\ldots\ in the font corresponding to
[style]~$C{\uparrow}$''. What we are doing is to increase~$q$. The amended
value of~$u$ will be subsequently used in Rule~18c to position the superscript.
\end{document}
Here’s the output:

Let us also magnify the crucial portion:

Of course, this solution could have drawbacks too: changing in this way a parameter that operate at so low a level of TeX might cause side-effects which are not evident at first sight.