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I have always thought that $\,\sin(x)^2$ was equivalent to $\,\sin^2(x)\,$ and not $\,\sin\left(x^2\right)$.
However, on a recent calc III midterm I did not receive credit for a problem as I wrote $\,\big(\!\sin(x)\big)^2$ as $\,\sin(x)^2$ and the prof said that $\,\sin(x)^2=\sin(x^2)$.
Does anyone know the answer to this ambiguity?

Angelo
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cold_snow_2006
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  • Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [meta], or in [chat]. Comments continuing discussion may be removed. – Xander Henderson Mar 19 '23 at 00:40

1 Answers1

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There is no "F" in "PEMDAS". So where F, function application, lies in the order of operations is left open, and $\sin(x)^2$ is thus ambiguous.

Now the fact that you've reported the question as using the notation "$\sin^7(4x-3y)$" means that the best policy would have been to use the same convention, and written "...$\sin^6(4x-3y)$...", keeping the exponent on the "sin".

If I had been grading your paper personally, I would have seen that you performed the other parts of the differentiation correctly, guessed that you understood how to compute partials and use the chain rule, wondered out loud "why did they introduce this unnecessary ambiguity when they've got the unambiguous notation right in front of them?", wondered what you meant, worried that you specifically used a different notation because you actually did have a misunderstanding of how it all worked, and thus felt you needed different notation to denote $\sin(x^2),$ and gotten a bit cranky. And I'm not sure what my point allocation would have been - maybe take 3 off out of 10, with a "Bad Notation" comment, but hard to say.

ryang
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JonathanZ
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    I very strongly disagree with your statement that the notation $\sin(x)^2$ is ambiguous. In fact, I would even argue that $\sin^2(x)$ (or $\sin^2 x$) is more ambiguous, as $f^n$ often means $f\circ f \circ f \circ \dotsb \circ f$ ($n$ times). The most unambiguous notation is to use more parentheses, e.g. $\bigl( \sin(x) \bigr)^2$, but I would never deduct points from an answer like $\sin(x)^2$, and would seriously question any instructor who did. – Xander Henderson Mar 19 '23 at 00:43
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    @XanderHenderson - So if you saw $\sin(4x-3y)^6$ in a mathematical paper, you would assume the author wanted you to evaluate the $\sin$ first, and then raise it to the power? Because that's the actual formula that this question was based on. And while I certainly can't question how you would interpret it, that's not how I would interpret it. – JonathanZ Mar 19 '23 at 01:11
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    I absolutely would, yes. Because that is how it works for all the other functions we work with. $f(4x-3y)^6 = [f(4x-3y)]^6$. The fact that we often omit the parentheses in trigonometric and logarithmic functions causes no end of confusion. I agree that the notation is potentially slightly ambiguous to a student, but anyone telling a student that $\sin(x)^2 = \sin(x^2)$ is doing a huge disservice to those students. – Xander Henderson Mar 19 '23 at 01:16
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    Also, referencing your statement that "there is no F in PEMDAS", I might assert that $f($ acts as an opening grouping symbol, and the closing $)$ finishes the job. So evaluating the function is at the priority of the G of "G/E/MD/AS" (G for "grouping"), and therefore takes priority over exponentiation. – Xander Henderson Mar 19 '23 at 01:23
  • Yup, we often omit the parentheses on $\sin$ unlike other function. And for a generic function there's the confusion you mention between raising to a power and multiple composition for $f^n()$, for $\sin$, and other trig functions, students have had "it means raise to the power" drilled into their heads since high school. Trig functions have their own conventions - it's a fact that we shouldn't ignore. And I def. wouldn't tell a student who wrote $\sin(x)^2$ that it was equal to $\sin(x^2)$ - I'd tell them "Don't do that". – JonathanZ Mar 19 '23 at 02:05
  • And I like what you observed about how $f($ acts to group, and that is reinforced when I realized $\sin:( x )^2$ would strike me differently than $\sin( x )^2$ - the bit of space between the function and the open paren in the first version acts like a top-level "break here" operator, in contrast to the group effect when they are contiguous. Regardless, I think that $\sin(x)^2$ is a collision of conventions and notation, and a student finishing their calculus class still thinking that it's perfectly fine notation is being done a disservice. – JonathanZ Mar 19 '23 at 02:13
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    @ryang - yeah, xander seems to have some strong beliefs about this, and I don't think I'm going to convince him otherwise. :-) Me, I think that it's ambiguous notation, and thus bad notation, and would definitely point that out to a student. In the end, it's Lubin who has the best answer in that thread: "The notation is a mess, and we’re stuck with it for purely historical reasons.". – JonathanZ Mar 19 '23 at 21:32
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    I deleted my previous comment, because it now contains a broken URL (the linked conversation having just been deleted by the community!). Have restored it instead to my answer on the same page, and have just upvoted your Answer (had neglected to do so earlier) to cushion this Closed page against deletion. – ryang Mar 20 '23 at 09:20