When do I write $\sin(x)$ and when $\sin x$? I sometimes see $\sin x$ and sometimes $\sin(x)$. Are the parenteses needed since the sine is a function or is it more an operator that can be premultiplied to the variable? Or are people just lazy?
 A: There's no mathematical difference in when to write parentheses or not, as long as there is no doubt how much of the thing that follows "$\sin$" is part of the argument.
Part of the syntactic role of parentheses is to make clear that the thing to the left of them is actually a function rather than something rather than something to be multiplied. The need for this is greater when the name of the function is just a letter ("$f$" or "$g$" could also conceivably be used as names of constants, for example), but on the other hand "$\sin$" is so unambiguously a function that we usually don't need parentheses to remind the reader that that's what it is.
... except in situations like $\sin(t+1)$ where "$\sin t + 1$" would have meant $(\sin t)+1$.
Omitting the parentheses in unambiguous cases makes the expression slightly easier to read at a glance then there are many other levels of parentheses around.
A: Parentheses make the expression clearer for expressions like $\sin (xy)$, if you write $\sin xy$, then it may mean  $(\sin x) \cdot y$. But for only  $\sin (x)$ it is enough to write $\sin x$. If there is some possibility of ambiguity, then it is better to use parentheses.
A: Note the small skip after the math operators
$$\sin x$$
vs.
$$\sin(x)$$
Thus parentheses are not necessarily needed, even for $\sin xy$. Similar, as the gap around the add operator $+$, this indicates that it has a higher precedence than the multiplication (cf. $a+xy$).
Sure, there is a tradeoff between ultimate safety and easy readability.
For more complicated equations, one may appreciate less parentheses
$$\exp(2(\cos x + i\sin xy))$$
$$\exp(2(\cos(x) + i\sin(xy)))$$
On blackboard (or bad formatted latex), parentheses are needed for clarity
$$sin x$$
$$sin(x)$$
Parenthesis are also needed for custom user functions, which don't switch font or lack the small skip:
$$f(x)$$
$${\rm Sin}x$$
BTW, $f(x)$ might also be interpreted as $f\cdot(x)$.
