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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?

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When it's not confusing, write $\sin x$. When it might confusing, write $\sin(x)$. When you're not sure, write $\sin(x)$. –  Asaf Karagila Oct 13 '12 at 17:28
    
It is definitely not "multiplying $x$". The issue maybe more when you have stuff like $\sin x^2$. It might not be clear if it is $(\sin x)^2$ or $\sin(x^2)$. Usually from the context this kind of issue is not relevant, it's clear, but if you don't feel comfortable just put the brackets. For instance, when one writes $2 \sin x \cos x = \sin 2x$, it is clear that it means $2 \sin(x) \cos(x) = \sin (2x)$. –  Patrick Da Silva Oct 13 '12 at 17:33
    
$\left(\sin(x)\right)^2$ is often denoted as $\sin^2 x$, which I would rather read as $(\sin \circ \sin)(x)$, but I guess that is another story. –  queueoverflow Oct 13 '12 at 17:35
    
@queueoverflow: There's no helping that power notation applied to function names can mean either repetition of function composition, or repetition of the pointwise multiplication operator. Both of these meanings are respectable and meaningful things to do. In the cases of positive powers of trig functions it happens that pointwise arithmetic is so much more common than repeated composition that this interpretation takes precedence, but the situation is exactly the opposite for $\sin^{-1}$. –  Henning Makholm Oct 13 '12 at 17:44
    
@queueoverflow So would you rather write it as $\sin x^2$ ? From my experience, (weak) students don't write brackets most of the times, and then the $\sin^2(x)$ leads to much less confusion... –  N. S. Oct 13 '12 at 18:05
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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.

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Parentheses make the expression clearer for the expressions like $\sin (xy)$, if you write $\sin xy$, then it may mean $(\sin x).y$. But for only $\sin (x)$ it is enough to write $\sin x$. If there is some possiblilty of ambiguity then it is better to use parentheses.

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Hmm... if you want $(\sin x)y$, then you'd better write out those parentheses. There are too many situations where one needs to speak about things like $\sin 2\pi\nu t$ for it to make sense that the invisible function application would bind stronger than invisible multiplication. –  Henning Makholm Oct 13 '12 at 17:59
    
Or write $y \sin x$ ;) –  N. S. Oct 13 '12 at 18:11
    
In addition to clarity for human readers, parentheses may also be required for computers. Who has tried sin x for Wolfram Alpha? –  GEdgar Oct 13 '12 at 18:37
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