Symbols:A/Arcsine/arcsin
Arcsine
- $\arcsin$
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 the arcsine of $x$.
Thus:
- The domain of arcsine is $\closedint {-1} 1$
- The image of arcsine is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arcsine function is $\arcsin$.
The $\LaTeX$ code for \(\arcsin\) is \arcsin .
Also denoted as
asin
- $\operatorname {asin}$
A variant symbol used to denote the arcsine function is $\operatorname {asin}$.
The $\LaTeX$ code for \(\operatorname {asin}\) is \operatorname {asin} .
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse trigonometric functions (antitrigonometric functions)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): arccos, arcsin, arctan, etc.${}$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse trigonometric functions (antitrigonometric functions)