Definition:Inverse Sine/Real/Arcsine
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Definition
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 $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Also denoted as
In computer software packages, the notation $\operatorname {asin}$ or $\operatorname {asn}$ can sometimes be found.
Caution
There exists the a 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 an 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 $\arcsin$, although this can also be confusing due to the visual similarity between that and the lowercase $\text s$.
Also see
- Results about inverse sine can be found here.
Other inverse trigonometrical ratios
- Definition:Arccosine
- Definition:Arctangent
- Definition:Arccotangent
- Definition:Arcsecant
- Definition:Arccosecant
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Principal Values for Inverse Trigonometrical Functions
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.5 \ (3)$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 23$: Restriction of a Mapping
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: inverse trigonometric function