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1st way:

One way to differentiate them would be the following:

The symbols for the inverse functions differ from the symbols for the inverse relations: the names of the functions are capitalized. The inverse functions appear as follows: Arcsine, Arccosine, Arctangent, Arccosecant, Arcsecant, and Arccotangent

So, relations are written non-capitalized ($\arcsin(x)$), and functions are written capitalized ($\text{Arcsin(x)}$).

2nd way:

Another way of writing the inverse trig relations/functions is like this: $\sin^{-1}(x)$. How can we differentiate between inverse trig relations and functions if it is written in this way? In other words, if I find $\sin^{-1}(x)$ written down on a piece of paper, what will I interpret it as: a function or a relation?

Questions:

  1. How can I differentiate $\sin^{-1}(x)$ as a function and $\sin^{-1}(x)$ as a relation?
  2. If I find $\sin^{-1}(\frac{1}{2})$ written down on a piece of paper, will I write $\sin^{-1}(\frac{1}{2})=30^{\circ}$ or $\sin^{-1}(\frac{1}{2})=150^{\circ}$?
tryingtobeastoic
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  • Both is fine, and each has advantages. – Aaa Lol_dude Jan 21 '22 at 05:11
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  • Context. 2. While IMO Arcsin clearly indicates the principal angle, I generally read arcsin too as the principal angle anyway (i.e., as a function rather than a relation). It would be a bad idea to assume that arcsin is automatically a relation, unless the text explicitly distinguishes between Arcsin and arcsin. 3. The same issue crops up in complex analysis for log vs. Log (ln vs. Ln).
    • – ryang Jan 21 '22 at 05:12
  • What is the difference in notation for Ln(z) vs. ln(z) – ryang Jan 21 '22 at 05:19
  • @ryang Hmmm. I edited the question slightly. – tryingtobeastoic Jan 21 '22 at 05:20
  • @ryang I see. What would you say about Q1? – tryingtobeastoic Jan 21 '22 at 05:35
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    For Q2, I don't think that $\sin^{-1}(\frac{1}{2})$ is ever straight-up interpreted as $150^{\circ}.$ If I'd like to list the entire collection of possibilites, I'd just say that $x$ is such that $\sin x=0.5$ or that $x\in{x\vert\sin x=0.5 }.$ I understand $150^{\circ}$ to be $\pi-\sin^{-1}(\frac{1}{2}).$ In any case, I think sin$^{-1}$ is just as distinct (or not effectively distinct, haha) from Sin$^{-1}$ as arcsin is from Arcsin. – ryang Jan 21 '22 at 05:35
  • https://www.dummies.com/article/academics-the-arts/math/trigonometry/how-to-distinguish-between-trigonometry-functions-and-relations-186918 – tryingtobeastoic Jan 21 '22 at 05:41
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    As a (very) minor criticism of the posting, it is a bad idea to use the verb differentiate here. In the Pre-Calculus world, you wouldn't have any way of knowing that the verb differentiate usually refers to taking derivatives (in Calculus). A synonym would be the verb distinguish. When I first read your posting, the verb differentiate threw me. It wasn't until I completed the posting and noticed that there was no Real-Analysis tag that I was able to reverse engineer what you intended by the verb differentiate. – user2661923 Jan 21 '22 at 07:08