Definition:Inverse Sine/Real/Arcsine

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Definition

Arcsine Function

From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

From Sine of Half-Integer Multiple of Pi:

$\map \sin {-\dfrac {\pi} 2} = -1$

and:

$\sin \dfrac {\pi} 2 = 1$


Therefore, let $g: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \to \closedint {-1} 1$ be the restriction of $\sin x$ to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.

Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$.


This function is called arcsine of $x$ and is written $\arcsin x$.


Thus:

The domain of $\arcsin x$ is $\closedint {-1} 1$
The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.


Terminology

There exists the popular but misleading notation $\sin^{-1} x$, which is supposed to denote the inverse sine function.

However, note that as $\sin x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have a well-defined inverse.

The $\arcsin$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\sin^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname {Sin}^{-1}$ (with a capital $\text S$) is taken to mean the same as $\arcsin$.

However, this can also be confusing due to the visual similarity between that and the lowercase $\text s$.


In computer software packages, the notation $\operatorname {asin}$ or $\operatorname {asn}$ can sometimes be found.

Some sources hyphenate: arc-sine.


Also see

  • Results about inverse sine can be found here.


Other inverse trigonometrical ratios


Sources

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