# What is this expression means? $\sin^{-2}x$

I know this is very silly question but I didn't know how to research it in google. Please bare with me on this one. I have two thoughts:

$$\sin^{-2}x = \frac{1}{\sin^2x}$$

I think this cannot be possible. Or is it:

$$\sin^{-2}x = \arcsin^2x$$

I didn't make up this question, I mean kinda. I am developing an android application which solves mathematical equations for demonstration purposes. I was brainstorming what the user can input, I came up with this expression.

• "I think this cannot be possible" Why? – Michael Galuza Aug 8 '15 at 17:00
• Could be both. It's bad notation. You have to guess by context. – tst Aug 8 '15 at 17:01
• The first one is correct. But do answer Michael's question – Shailesh Aug 8 '15 at 17:01
• Because if this can be possible, then sin powered to -1 wouldn't be arcsin. I think that way. – Bora Aug 8 '15 at 17:02
• @MichaelGaluza: Some people use $\sin^{-1} x$ to mean $\arcsin x$, that's the point. – Javier Aug 8 '15 at 17:31

## 2 Answers

For a computer application, I would suggest you avoid all ambiguity by forcing the user to type $$(\sin x)^2$$ for $\sin^2(x)$, and $$\arcsin x$$ for the inverse function. Don't allow $\sin^n (x)$ as it is abusive shorthand which has done more to confuse people than help.

However, it's also worth noting that Wolfram Alpha interprets $\sin^{-2}(x) = \frac{1}{(\sin x)^2}$, whereas $\sin^{-1}(x) = \arcsin x$. More generally, it appears that $n = 1$ is a special case in $\sin^n(x)$ and otherwise it is interpreted as $(\sin x)^n$.

The reason $\sin^n(x)$ is bad notation for $(\sin x)^n$ is that in most contexts, $f^n(x)$ means the $n$th iterate of $f$. In particular, the $-1$th iterate of an invertible function is the inverse, so $\sin^{-1}$ would actually be a natural way to denote $\arcsin$.

On the other hand, the problem with $\sin^{-1}(x)$ for $\arcsin$ is of course that it conflicts with the hugely widespread notation $\sin^2(x)$. You really can't win.

• +1, though I disagree that $f^n(x)$ means the nth iterate in every other context. In abstract algebra it's a common notion (as usually $f^n$ refers to $\underbrace{f\cdot f\cdot f\cdots f}_{n\text{ dots}}$, and the operation $\cdot$ is simply composition), but not in other circumstances. I've seen $\ln^n(x)$, for example, and $J_n^m(x)$ very often means the Bessel function of order $n$ raised to the $m$th power. The alternative notation for the nth iterate of $f$, $f^{\circ\ n}(x)$, isn't that universal either however. It's best just to state the definition of $f^n$ in context. – user3002473 Aug 8 '15 at 17:56
• $\sin(x)^n$ please. – dbanet Aug 8 '15 at 18:05
• @user3002473 That's a good point. But $f^{-1}$ very universally means inverse, I think. Whenever I see $f^n$ at least for a general function $f$ I assume it means iterate. Anyway I'll update my answer to make the wording a little less strong. – 6005 Aug 8 '15 at 18:21
• @6005 Yeah true, I've never heard of a context where $f^{-1} = 1/f$. Historically, that makes wonder if the notation $f^n$ started in abstract algebra so that $f^{-1}$ clearly indicates the inverse, and then other fields borrowed from that. – user3002473 Aug 8 '15 at 18:26

(1) If you found this in a book, paper, or web page, without an explanation, that is a bad book, paper or web page.

(2) Do not use this notation yourself.