About the use of curly braces

Note that the braces are not just not required, they should not be used as \cos does not take an argument the braces form a group and force the term to lose any special math class that it would otherwise have.

\cos is a "\mathop" operator and ( is a "\mathopen" but a term surrounded by {..} is always a "\mathord". TeX uses different space between a \mathop and a \mathopen than between a \mathop an a \mathord (because log(x) does not need any space but log x does).







which produces

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operators like \cos or \sin don't require curly braces, because they don't take arguments; they're simply substitutions.

any command that takes an argument is best fed that argument in braces. even if the argument is a single, unexpandable token (such as the letter "x"), where braces are optional, it's not improper to provide the braces. in fact, it's probably a good idea to get in the habit of using them, since what may look like a single token (it will always begin with a backslash) may expand to more than one token and cause problems.

(the final example in the question would be improved by the addition of a thin space: r\sin \theta \, d\phi.)