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

I have worked a bit on trigonometry today, and something strikes me as inconsistent.

In the book, the notation for the inverse sine function is $\sin^{-1}$, but the same notation is also used in $\sin^2$ meaning $(\sin x)^2$.

Are there any alternate notations which avoid this ambiguity?

Some examples:

$\sin30^\circ = 0.5$
$\sin^2 30^\circ = (\sin 30^\circ)^2 = 0.25$

However, inverse sine does not work that way:
$\sin^{-1} 30^\circ \ne (\sin 30^\circ)^{-1}$
$\sin^{-1} 30^\circ = $ Error or complex number?
$(\sin 30^\circ)^{-1} = 2$

The potential confusion only gets worse if you use radians, as they are in the range [-1, 1] for [-57°, 57°].:

$\sin^{-1} 0.524 = 0.551$
$(\sin 0.524)^{-1} = 2$

And what if you want both at the same time? You are forced to use parentheses, thus breaking any consistency: $(\sin^{-1} x)^2$

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marked as duplicate by J. M., Hans Lundmark, Asaf Karagila, Gerry Myerson, Zev Chonoles Feb 19 '12 at 13:48

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.

$\arcsin$ instead of $\sin^{-1}$ – john w. Feb 13 '12 at 3:44
Indeed it is unfortunate notation. And the better ($\arcsin$) notation is gradually gettiing less common, $\sin^{-1}$ is easier to squeeze on a calculator keyboard. – André Nicolas Feb 13 '12 at 4:14
It's interesting that in Serbia we never used $sin^{-1}$ and I got kinda confused when I seen that notation everywhere around the world. We learned $arc$s only. So as people already said, $arcsin$ could solve your problem. (It's also interesting we use $tg$ for $tan$ and $ctg$ for $cot$). – Lazar Ljubenović Feb 13 '12 at 9:41

2 Answers 2

up vote 0 down vote accepted

The inverse trigonometric functions are also called arcfunctions, as they return the unit circle arc length (in radians) for a particular value of sine, cosine, etc.

$$\sin^{-1}(x)=\arcsin x$$ $$\cos^{-1}(x)=\arccos x$$

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Is asin acceptable an abbreviation for arcsin? It is used a lot in programming languages. – CheeseSucker Feb 13 '12 at 19:47
@CheeseSucker: I've seen those labels on the buttons on HP calculators, but I don't know that I've ever seen them in mathematical writing. – Isaac Feb 13 '12 at 22:39

Carl Gauss also objected to this particular notational inconsistency, so you're in good company. He proposed that $\sin^2(x)$ ought to mean $\sin(\sin(x))$, whereas $(\sin x)^2$ should be written in that way.

But the notation in the book you mention is standard. I don't think there's any standard alternative.

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