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In the notation $\sin(x)^2 $

Does this equal $\sin(x^2) $ or $(\sin(x))^2 $ ?

I'm sorry this is such a simple question but Google is unhelpful. There are plenty of sources illustrating $\sin^2(x) $ = $(\sin(x))^2 $ but nothing about $\sin(x)^2 $

Edit

According to the answers in my math book(Calculus of a Single Variable 10e, Ron Larson & Bruce Edwards), it seems as if it equates $\sin(x)^2$ as $\sin(x^2)$ This obviously isn't proof of the notation but it would make some sense when considering $\sin(x+y)^2$ or $\sin(2x)^2$ as opposed to a single variable. I find this notation very confusing and better stated explicitly such as: $\sin((x+y)^2)$ or $\sin((2x)^2)$.

I'm still not happy with this answer and would appreciate if anyone could reference evidence to one side or the other.

Michael Hardy
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Sam Sabin
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    It can mean either, depending on the author. This is precisely why the notation $\sin^2x$ exists. – Ben Grossmann Oct 04 '15 at 23:53
  • That is, there is no universally accepted "order of operations" that decides priority between function composition and exponentiation. – Ben Grossmann Oct 04 '15 at 23:55
  • How can it mean either? This is important because I'm doing differentiation problems with the chain rule and this significantly changes the answer. Which is more commonly accepted among mathematicians? – Sam Sabin Oct 04 '15 at 23:58
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    @Omnomnomnom: I would agree there is no universally accepted order of operations, but I suspect most mathematicians would read $\sin(x)^2=(\sin(x))^2$ but $\sin x^2 = \sin(x^2)$. Given that most of them read $\sin^2 x = (\sin(x))^2$ but $\sin^{-1} x \not = \frac{1}{\sin(x)}$, their opinions can be expected to be hopelessly inconsistent. – Henry Oct 05 '15 at 00:02
  • it would be helpful if you can post the question. It can mean either because this is a bad notation. – SamC Oct 05 '15 at 00:02
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    It is ambiguous, but it "looks" like $(\sin{x})^2$ to me. Or at least I would assume it was until proven otherwise: $\sin^2{x}$ is many times more common than $\sin{(x^2)}$. – Chappers Oct 05 '15 at 00:02
  • I agree this is a bad notation but this is the notation it uses in my math book. – Sam Sabin Oct 05 '15 at 00:03
  • As others said, if i saw this in the text ill read this as $\sin^2(x)$. – SamC Oct 05 '15 at 00:05
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    I like the top answer here: http://math.stackexchange.com/questions/932903/ambiguity-of-notation-sinx2?rq=1 – zahbaz Oct 05 '15 at 00:11
  • How about $sin(2x)^2=sin(4x^2)$.... – NoChance Oct 05 '15 at 00:44
  • I think the first comment on the first answer illustrates what I was thinking. Adding variables in the parentheses could possibly justify using that notation if you were just trying to make it shorter even though it is still confusing. – Sam Sabin Oct 05 '15 at 00:52
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    According to the answers to some of the questions in my book, it seems as if $\sin(x)^2$ = $\sin(x^2)$ in my book. I think this is for the reason Emmad Kareem comes up with. – Sam Sabin Oct 05 '15 at 00:54
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    I would never write $\sin(x)^2$, precisely because of the annoying ambiguity. $(\sin x)^2$ and $\sin(x^2)$ are clear, and it has become universal to write $\sin^2 x$ for the former, even though it makes sense that that notation ought to mean $\sin\sin x$. ${}\qquad{}$ – Michael Hardy Oct 05 '15 at 01:11
  • Where (book? pages? context?) did you find the notation $\sin(x)^2$? –  Oct 05 '15 at 01:26
  • Calculus of a Single Variable 10e, Ron Larson & Bruce Edwards, section 2.4 Exercises(pg.136) it has examples such as 47. $y = \sin(πx)^2$, 48. $y = \cos(1-2x)^2$, and 60. $y = 3x - 5\cos(πx)^2)$. There are numerous other examples on this page and the answers at the back of the book correspond with using $\sin(x)^2$ or $\sin(ax)^2$ as $\sin(x^2)$ and $\sin((ax)^2)$. The instructions for these questions is to find the derivative(using the chain rule). – Sam Sabin Oct 06 '15 at 22:07
  • I totally agree with @MichaelHardy. – ryang Mar 21 '23 at 07:56

1 Answers1

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$$\sin^2(x)=(\sin x)^2=\sin (x)^2\neq \sin(x^2)$$

yankeefan11
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