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$\arcsin$ written as $\sin^{-1}(x)$

When I learnt the trig identity $\sin^2\theta + \cos^2\theta \equiv 1$, I learnt that $\sin^2\theta = (\sin\theta)^2$.

So why isn't $\sin^{-1}\theta = (\sin\theta)^{-1} = \dfrac{1}{\sin\theta}$?

Because $\csc\theta = \dfrac{1}{\sin\theta} $, but $\csc\theta \ne \sin^{-1}\theta$

How can these two same notations, just with different numbers, mean different things?

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marked as duplicate by J. M., Blue, Henning Makholm, Ross Millikan, Zev Chonoles Jun 23 '12 at 2:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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In the good old days, the notation $\arcsin x$ was the common one. The greater current popularity of $\sin^{-1} x$ is probably due to calculators: $\arcsin$ does no fit nicely into the cramped space above the $\sin$ buton. –  André Nicolas Jun 22 '12 at 23:00

5 Answers 5

It is a notational thing, there is nothing mathematically wrong here.

We denote the inverse of a function $f$ by $f^{-1}$. So when you see sin$^{-1}(x)$, this means "the inverse sin function", not the reciprocal.

But also sin$^2 (x)$ is just notation for $(\text{sin}(x))^2$, there is no mathematical reason why they are equal, it is by definition.

I know it is slightly confusing but it is notation that has stuck through time...

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It's only a matter of convention. $f^{-1}$ is used to mean the inverse function of $f$, pretty much always. Only with $\cos$, $\sin$ and other trigonometric functions we usually multiply those together, or square or cube them. That's why it becomes useful to write $\cos^2 x$ as a shortcut for $(\cos x)^2$, for example. But it's just notation (that can be confusing at times).

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why can't $cos^2X$ be written as $cos(X)^2$ (which is different to $cosX^2 == cos(X^2)$)? –  Jonathan. Jun 23 '12 at 11:47
    
@Jonathan. Sure it can, but $\cos^2 x$ is shorter, and maybe $\cos (x)^2$ can be easily mixed up with $\cos (x^2)$. –  talmid Jun 23 '12 at 16:05
    
+1 for "it's just a convention". You can use whatever notation you care to use for whatever you care to use it for - you'll only have problems if you try to communicate with someone else. The Nobel Prize-winning physicist Richard Feynman developed his own notation for trigonometric functions because, as a self-taught mathematician, he did not initially know the standard notation. Sadly, I am not Richard Feynman, I am not a self-taught mathematicians who conjured trigonometry from thin air while in high school, and thus it's a good idea for me, at least, to use the standard notation for trig. –  Bob Jarvis Sep 16 at 3:14

I seem to recall reading that Gauss said that the notation $\sin^2\theta$ ought to mean $\sin(\sin(\theta))$, and similarly for other powers. That would be consistent with using $\sin^{-1}$ to mean the inverse function.

So $\sin^{-1}$ means the inverse function because $\sin^3\theta$ ought to mean $\sin\sin\sin\theta$.

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Because $\sin^{-1}$ is thought of more as the inverse function. So let's say $f = \sin$. Then the inverse function if $f^{-1} = \sin^{-1}$. Also, $\sin^{-1}$ can be also written as $\arcsin$.

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This universally observed convention is just an example of overloading of notation, where the meaning of notation depends upon context.

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