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Sine is a function and functions are all written as $f(x)$ where $f$ is the 'name' of the function. We never see any textbook using $f\ x$ to denote a function of $x$.

And also, using $\sin(x)$ saves us from blunders like $\sin x^2$ which can be interpreted as either $\sin(x^2)$ or $\sin(x)^2$. Even writing $\sin(x)^2$ as $\sin^2 x$, is not a very good notation because, $f^2(x)$ generally means $f\circ f(x)$. And $\sin^2 x$ can be interpreted as $\sin(\sin(x))$ which is not equal to $\sin(x)^2$. This contributes to things like $\sin^{-1} x$, which I think doesn't make any sense. I mean, what does $\sin^{-1}x$ supposed to mean? If $\sin^2 x$ is $(\sin x)^2$, then, $$\sin^{-1}x=(\sin x)^{-1}=\frac{1}{\sin x}\qquad\text{or}\qquad\sin^{-1}x=\frac{1}{\sin}(x)$$ $\displaystyle\frac{1}{\sin}(x)$ feels so strange. But $\sin^{-1} x$ is supposed to mean $\arcsin(x)$. Same things happen with all trigonometric functions. Shouldn't $\sin x$ be inaccurate way of denoting $\sin(x)$? Then why many, many books write $\sin(x)$ as $\sin x$? Isn't it plain wrong?

zar
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sigsegv
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    It is convention. – copper.hat Oct 27 '16 at 12:29
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    Nothing is wrong as long as it doesn't create ambiguity. I have no objection to $\sin x^2$, which I'll never interpret as $\sin^2x$. And to me $f^2(x)$ is the square, not the second iterate, for which I would prefer $f^{(2)}$. Anyway, this can also denote the second derivative and a warning should be givien to the reader if context is insufficient :( –  Oct 27 '16 at 12:30
  • there is no doubt with $sin(x) $. – hamam_Abdallah Oct 27 '16 at 12:35
  • Nobody will object to $ax^2+bx+c$ though one should write $(ax^2)+(bx)+c$. The same holds for $\sin x$. $f x$ isn't used because of the possible confusion with a product. Sometimes, the application of a function to a variable is denoted $f.x$. –  Oct 27 '16 at 12:38
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    @YvesDaoust I recommend against writing $f^{(n)}$ for the $n^{\text{th}}$ iterate, since that more commonly denotes the $n^{\text{th}}$ derivative. In dynamics, $f^n$ is quite common notation for the $n^{\text{th}}$ iterate, though I have seen $f^{\circ n}$ used to avoid ambiguity. – Mark McClure Oct 27 '16 at 12:40
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    @YvesDaoust But $f^{(2)}(x)$ may mean the second derivative of the function, i.e $f''(x)$. The second iterate is written as $f^2(x)$. This creates even more ambiguity. – sigsegv Oct 27 '16 at 12:43
  • This question is basically a textbook case of all the exceptions that $\sin$ and $\cos$ are given for the otherwise generally accepted rules of notation for functions. In particular, the fact that $\sin^{-1}$ and $\sin^2$ denote inconsistent notations with each other is the reason for preferring the use of $\operatorname{arcsin}$ for the inverse function. – Dustan Levenstein Oct 27 '16 at 12:47
  • @MarkMcClure: I already mentioned that. –  Oct 27 '16 at 12:48
  • @atayana: I already mentioned that. Is it so unreadable ? –  Oct 27 '16 at 12:49
  • It doesn't matter, once people understand what you mean. But it is a bit strange that $\sin^{-1} x$ sometimes means $\arcsin$ and $\sin^2 x$ usually means $(\sin x)^2$, not $\sin \sin x$. – Peter Franek Oct 27 '16 at 12:50
  • @PeterFranek: I agree with you. I would interpret $f^2(x)$ as a square but $f^{-1}(x)$ as an inverse $f^{-1}(f(x))=x$, never as a reciprocal $1/f(x)$. But $f^{-2}(x)$ wouldn't denote the iterated inverse $f^{-1}(f^{-1}(x))$ but the squared reciprocal $1/f^2(x)$ ! –  Oct 27 '16 at 12:55
  • @YvesDaoust Uhm, you explicitly stated in your first comment that you prefer $f^{(2)}$ for the second iterate. I advise against that. – Mark McClure Oct 27 '16 at 13:17
  • @MarkMcClure: I also explicitly stated that there is ambiguity with the second derivative and the reader must be warned, didn't I ? –  Oct 27 '16 at 13:29
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    @YvesDaoust Yes, you certainly did. Why you explicitly prefer $f^{(2)}$ for the second iterate, I don't know. As someone who works in dynamics, I find that odd. That is my point, which I share with at least one other observer, and that is all. :) – Mark McClure Oct 27 '16 at 13:44
  • @MarkMcClure: the comment by atayana is even less relevant. –  Oct 27 '16 at 13:46

2 Answers2

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It is not wrong per se.

You can choose to write $f\,x$ if $f$ is a function which is applying to $x$. But you always need to be careful that it doesn't create ambiguity (because all your arguments are recevables).

For instance, authors often specify when they are writing $f^{-1}$ if it means $\frac 1{f(x)}$ or $f^{-1}(x)$ where $f^{-1}$ is such that $f^{-1}\circ f=f\circ f^{-1}=\text{id}$.

E. Joseph
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I usually see $\sin x$ being used in most books, but I often prefer $sin(x)$ instead. Most authors will eventually resort to write $sin(f(x))$ with parenthesis whenever they need a $f(x)$ other than $x$ in the argument of $sin$, so I chose to always use parenthesis.

Also, I do enjoy using more $sin^2(x)$ instead of $sin(x)^2$ to denote $sin(x)\cdot sin(x)$. In my experience, the second power of $sin(x)$ is much more common (in the contexts I work/lecture) than $(sin(sin(x))$ so, usually, there's no chance to take one for another.

Also, I've seen $sin^{-1}(x)$ and $sin(x)^{-1}$ to denote $arcsin(x)$, even though $sin(x)^{-1}$ could also be mistaken for $\dfrac{1}{sin(x)}$. I guess it all comes down to define on your text/lecture what notation will be used for each context. That is always the best way to settle things down.

One more thing: if one writes $f^{(2)}(x)$ to denote the second iterate of $f$, it could also be mistaken for the second derivative of $f$ (in the context of Differential Calculus). Some would resort to denote derivatives with Leibniz notation $\dfrac{d^2f}{dx^2}(x)$, but that sometimes demands too much notation for simple things. I conclude that there is no better way to write stuff; just make a convention on your paper/notes/lectures beforehand and stick to it.

Marra
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