Category:Arcsine Function
This category contains results about 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 $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Also see
Subcategories
This category has the following 4 subcategories, out of 4 total.
Pages in category "Arcsine Function"
The following 19 pages are in this category, out of 19 total.
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- Arccosecant of Reciprocal equals Arcsine
- Arcsine as Integral
- Arcsine Function in terms of Gaussian Hypergeometric Function
- Arcsine in terms of Arctangent
- Arcsine Logarithmic Formulation
- Arcsine of One is Half Pi
- Arcsine of Reciprocal equals Arccosecant
- Arcsine of Zero is Zero
- Arctangent in terms of Arcsine