# Isn't writing sin x instead of sin (x) wrong?

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?

• It is convention. – copper.hat Oct 27 '16 at 12:29
• 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 :( – Yves Daoust Oct 27 '16 at 12:30
• @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
• @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
• @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

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 $f^{-1}\circ f=f\circ f^{-1}=\text{id}$.

• But then, how will you say the difference between, function of $x$ and $f$ times $x$? – sigsegv Oct 27 '16 at 14:24
• @Depends on the context, but it is clear that $f\, x$ means $f(x)$ when you know that $f$ is a function. For instance $\sin$. – E. Joseph Oct 27 '16 at 14:25
• There are other authors who even write $xf$ instead of $f(x)$. – Xam Oct 27 '16 at 14:40
• In the context of $C^*$ algebras one can see even $f(x)=x(f)$. – A.Γ. Oct 27 '16 at 15:03

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.