# 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.

### A

- 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