Spacing of negative sign next to things like \sin

The extra bit of space inserted between the unary - symbol and the string sin is not unique to \sin: it applies to all objects with math status mathop ("math operator"), such as \det, \exp, \tan, etc. As the previous commenters have noted, this is very much a conscious design decision. If nothing else, this design decision makes is much less likely that "sin" will be mistaken for the variable name.

[In $-\sin(x)-\sin(x)$,] the spacing between the negative sign and the first "sin" seems to be just like the one between the minus and the second "sin". Well, perhaps a tiny bit smaller.

The amount of whitespace between the first - symbol (a unary operator) and "sin" is governed by \thinmuskip, which equals 3mu in most document classes I'm familiar with. (18mu=1em) The amount of whitespace that surrounds the second - symbol (a binary operator) is governed by the parameter \medmuskip, which equals 4mu (plus an amound of "glue") in all document classes I'm familiar with.

Let's simplify your example (the $-u-u$ is essentially the same as $-2-2$) and ask TeX to show us the math lists before it starts converting them to horizontal lists:

\documentclass{article}
\begin{document}

\showboxbreadth=20
\showboxdepth=1000
$-2-2 \showlists$. $-\sin(x)-\sin(x) \showlists$.

\end{document}

(you can of course choose other values than 20 and 1000, but if you choose too small values, you won't see the desired contents)

With this, you'll see the following in the log file:

### math mode entered at line 6
\mathbin
.\fam2 ^^@
\mathord
.\fam0 2
\mathbin
.\fam2 ^^@
\mathord
.\fam0 2

This is for the first equation. The \mathbin atoms are your minus signs and the \mathord atoms are the 2s.¹ Thanks to Marcel Krüger's comment, we know that the first \mathbin won't really stay a \mathbin, because TeX doesn't allow this at the beginning of a formula (I had forgotten this rule). It will act like a \mathord, as far as spacing goes—see below. So, the small space between unary minus and 2 is the space between two \mathord atoms. The log file will also show the math list built from the second equation:

\mathbin
.\fam2 ^^@
\mathop\nolimits
.\mathord
..\fam0 s
.\mathord
..\fam0 i
.\mathord
..\fam0 n
\mathopen
.\fam0 (
\mathord
.\fam1 x
\mathclose
.\fam0 )
\mathbin
.\fam2 ^^@
\mathop\nolimits
.\mathord
..\fam0 s
.\mathord
..\fam0 i
.\mathord
..\fam0 n
\mathopen
.\fam0 (
\mathord
.\fam1 x
\mathclose
.\fam0 )

As you can see, both minus signs have again been turned into \mathbin atoms (but the one at the beginning of the forumula wil be eventually treated like a \mathord, see above and Marcel Krüger's comment), precisely:

\mathbin
.\fam2 ^^@

The big difference with the first formula is that the \sin is a \mathop, not a \mathord as are the 2s and the us from your example. If you use bracing to make the first \sin(x) a subformula, it becomes wrapped in a \mathord and you get tight spacing (I'm not pronouncing myself on whether this is good typography):

\documentclass{article}
\begin{document}

\showboxbreadth=20
\showboxdepth=1000
$-2-2$. $-{\sin(x)}-\sin(x) \showlists$.

\end{document}

The first \sin(x) then produces this output in the log file:

### math mode entered at line 6
\mathbin
.\fam2 ^^@
\mathord
.\mathop\nolimits
..\mathord
...\fam0 s
..\mathord
...\fam0 i
..\mathord
...\fam0 n
.\mathopen
..\fam0 (
.\mathord
..\fam1 x
.\mathclose
..\fam0 )

The corresponding rendered output:

Now, thanks to Marcel Krüger's comment, we can explain why the space after the two minus signs starting the formulas are different in the initial example before the 2 and before the \sin: because the minus signs are \mathbin atoms occurring at the start of a formula, TeX actually treats them as \mathord atoms, so according to page 170 of the TeXbook (which is reproduced in this answer of egreg), the space after the first - in $-2-2$ is the space between two consecutive \mathord atoms, i.e., no space at all. On the other hand, the space after the first - in $-\sin(x)-\sin(x)$ is the space between a \mathord on the left and a \mathop on the right, namely a thin space (defined by \thinmuskip, 3mu in plain TeX).

If we wrap the first \sin(x)$ in braces, which makes it a \mathord, we again get the spacing between two consecutive \mathord atoms, i.e., no space at all, as with the 2s and the us.

---

Footnote

1. In case you wonder why the minus sign appears as an atom whose nucleus is \fam2 ^^@, here is the explanation. Math family 2 is used by TeX for “basic” math symbols (extensible ones like parentheses, braces, integral signs, \sum, \sqrt, etc., are taken from family 3 instead). The font that is assigned by default to this family is cmsy, whose encoding is called OMS and shown (currently on page 33) in encguide.pdf. As you can see in the following excerpt of cmsy10s' font table: the glyph in slot 0 of this font is the minus sign. And this is precisely the glyph (presented as \fam2 ^^@) in the math list we got from \showlists. Indeed, since the ASCII character with code 0 (aka NUL) is not in the range of printable characters, TeX displays it using its ^^ notation. In this notation, ^^@ represents the character with code 0 because the TeX-internal code for @ is 64, which lies between 64 and 127; therefore, TeX substracts 64 from this internal code, which gives 0. Had the TeX-internal code been between 0 and 63, TeX would have added 64 to the code instead. This rule is described in the TeXbook p. 45. In case you wonder, the “TeX-internal code table” a priori coincides with the ASCII code table for traditional TeX engines:

   TeX's internal code is based on the American Standard Code for Information Interchange, known popularly as “ASCII.”

   (quote from the TeXbook Appendix C, p. 367, which has a complete table of the encoding).

TeX's spacing rules are surprisingly simple, but make a reasonable job for a large class of expressions, sometimes you may need to help it out with some additional space such as \! (but not I think here).

The rules are based on the classification of individual atoms (ordinary, infix binary operator, or infix relation, prefix operator) and a matrix giving the space assigned between adjacent atoms of each class. There is no global analysis of the structure of the expression and no real understanding of its "meaning".

Firstly, as Mico noted in $-\sin x$ the - is (as you comment in the question) converted from a mathbin to a mathord so gets no extra space to the left, so the space between - and sin is the space between a mathord and a mathop. That is a thin space (slightly less than the space normally added around a mathbin in $a-b$). This space has to work for all mathord mathop situations, it does not treat - in any special way here, so the typical case might be $a \sin x$

$a\sin x$

$a{\sin} x$

$a{\sin x}$

In your case a is replaced by - so there is already better visual separation between the mathord and the mathop, but TeX doesn't "know" that.

The top is the default spacing (which is probably best for the largest set of cases) The second uses {\sin} so that is also a mathord and no additional space is added at all, as the expression is just three mathord.

The third is perhaps the one you want where the space to the left of the \mathop is suppressed by making \sin x a mathord, but it isn't such a good default spacing and it's not clearly an improvement even if a is replaced by - as in your original example.