Which one is better?
$$\arccos^2x^2\quad\quad \text{or}\quad\quad(\arccos x^2)^2$$
Are both notations acceptable?
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1A matter of taste. I prefer the more compact first version. – Peter Jan 09 '21 at 09:42
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1Both are fine. I often encounter the first one. – Wuestenfux Jan 09 '21 at 09:43
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3If I used the first version, I'd put parens around the x^2, for style reasons. – Hank Igoe Jan 09 '21 at 09:43
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Ok, I got it. thank you guys! – Etemon Jan 09 '21 at 09:47
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1I use both. I prefer the first, but sometimes there is a risk of confusing it with double derivative (which should be either $\arccos''$ or $\arccos^{(2)}$), or function composition. Parens around $x^2$ could also be used in both. – Jean-Claude Arbaut Jan 09 '21 at 09:49
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I remember the first term being used much often in my books – krenerd Jan 09 '21 at 09:52
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It's unfortunate that many people still use the first notation, it is awful. There are too many possible confusions as Jean-Claude pointed out. The second notation is much better. – Qi Zhu Jan 09 '21 at 09:55
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Make life simpler using brackets of parentheses. – Claude Leibovici Jan 09 '21 at 10:19
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I also would use $\arccos^2(x^2)$, see also https://math.stackexchange.com/questions/544360/should-i-use-parentheses-when-writing-log/544563#544563 – Michael Hoppe Jan 09 '21 at 14:11
1 Answers
2
Is $\arccos^2x$
- the inverse function of $\cos^2x$?
- the arccosine squared?
The first notation is therefore ambiguous. I always write $\cos^{-1}$ to mean $\arccos$, and so I would write the given expression as $$(\cos^{-1}x^2)^2$$
Parcly Taxel
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1Contemporary schools deprecate $\cos^{-1}$ as being ambiguous and imprecise. It is a multifunction with an unspecified principal branch. OTOH $\arccos$ is precisely and conventionally defined to mean "the branch of $\cos^{-1}$ whose image set is ${x \in \mathbb R: 0 \le x \le \pi}$". – Prime Mover Jan 09 '21 at 12:50
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@PrimeMover If $\log$ is also multivalued, why isn't it deprecated? – Parcly Taxel Jan 09 '21 at 12:58
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