Definition:Inverse Sine/Arcsine
Definition
Real Numbers
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 $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Complex Plane
The principal branch of the complex inverse sine function is defined as:
- $\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
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.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): arc cosine, arc sine, arc tangent, etc.${}$