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So in trigonometry, we have sin, secant (which is one over sin) and arcisn. Why is arcsin sometimes represented with sin^-1? sin^2 means sin to the second power, but sin^-1 explicitly does not mean sin to the negative first power, as that would be the secant, not the arcsin. Why this confusion in notation?

Git Gud
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Daniel B
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    In more advanced mathematics, the inverse function is denoted that way. If the function is $f$ then the inverse function is denoted $f^{-1}$. – GEdgar Jan 24 '15 at 18:38
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    In general, if $f(x)$ is a function, we write the inverse function as $f^{-1}(x)$. It's confusing notation at times, since we write $\sin^2(x)$ for $(\sin x)^2$, not $\sin(\sin(x))$. – Thomas Andrews Jan 24 '15 at 18:39
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    Because of this confusion, I recommend writing $\mathrm{arcsin}, x$ in your own writing. But of course you have to recognize $\sin^{-1} x$ when written by others. – GEdgar Jan 24 '15 at 18:40
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    The notation for inverse functions is highly inconsistent in maths. For instance $\log$ is used instead of $\exp^{-1}$, so if you feel the use $\sin^{-1}$ is confusing, you may use $\arcsin$ instead, or maybe write $(\sin(x))^2$ instead of $\sin^2(x)$, but be consistent with your use. Don't use both $\arcsin(x)$ and $\sin^{-1}(x)$. – Frank Vel Jan 24 '15 at 18:47
  • That's how you write inverse of a function. – AvZ Jan 24 '15 at 19:09
  • The worst thing is when someone writes $sin^{-2}$, very hard to tell what they mean. My friend linked me a derivation before with that in it. Fair bit of debating over that one – snulty Jan 24 '15 at 19:59
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    Also $\sec x=\frac{1}{\cos x}$ – snulty Jan 24 '15 at 20:03
  • Related: https://math.stackexchange.com/questions/30317/arcsin-written-as-sin-1x – Hans Lundmark Aug 30 '18 at 06:30
  • It's the secret advanced mathematician question: What does $\sin^{-1}(x)$ mean? – Lee Mosher Dec 15 '23 at 01:18
  • The situation is even worse in that the function denoted $\sin^{-1}$ isn't actually the inverse of $\sin$---that function fails the horizontal line test and so has no inverse. Instead it is the inverse of the restriction of $\sin$ to $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. – Travis Willse Dec 15 '23 at 01:23

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If we agree, that $f^{-1}$ means the inverse of $f$, we may obtain such strange notation. Let us observe, that $x^{-1}=1/x$ is not the inverse of $f(x)=x$, $x^{-3}$ is not the inverse of $x^3$ and so on.

Przemysław Scherwentke
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    The notation wouldn't have that inconsistency problem because in $f^{-1}$ the $-1$ is applied to the name $f$ of the function. In the case $x^{-1}$, the $x$ is not playing the role of the name of a function, but of the value $f(x)$ of the function called $f$ and defined as $f(x)=x$. To be precise $x^{-1}=(f(x))^{-1}$. So, $-1$, when applied as superscript means: The inverse function, when applied to the name of the function, or the reciprocal when applied to a value. – Pp.. Jan 24 '15 at 18:49
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    I prefer to use $\arcsin$, but actually, the notation that causes the confusion is rather $\sin^2(x)$ to mean $(\sin(x))^2$ instead of $\sin(\sin(x))$. But again, I also prefer to use $\sin^2(x)$ with the former meaning. After all, $\sin(\sin(x))$ is a less common expression to stumble upon. – Pp.. Jan 24 '15 at 18:53
  • @Pp.. Nevertheless, the notation $f^*$, as some people suggest, seems to be more resistant for students' mistakes. – Przemysław Scherwentke Jan 24 '15 at 18:54
  • @PrzemysławScherwentke But $f^*$ is already used for other purposes such as dual mapping… – Divide1918 Dec 15 '23 at 02:08
  • @Divide1918 If the students know the dual mapping, they do not have problems with $\operatorname{arc}\sin(x)$, I hope. – Przemysław Scherwentke Dec 16 '23 at 23:05