---
tag: 0B89D93YA1
kind: definition
name: Arcsine
msc: [33B10]
defines: 00Z1YBJJW8
uses: [0WB57D9488, 07BY47HG5S, 0C1VYVNV2N]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    locator: Principal values
---

## Definition

The arcsine is the function $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$ such that, for each $x \in [-1, 1]$, $\arcsin x$ is the unique $y \in [-\pi/2, \pi/2]$ with $\sin y = x$.

## Remarks

The sine is strictly increasing and continuous on $[-\pi/2, \pi/2]$, with range $[-1, 1]$. So $\arcsin$ is well defined and [[continuous|07BY47HG5S]].
For example, $\arcsin 0 = 0$, $\arcsin 1 = \pi/2$, and $\arcsin \frac{1}{2} = \pi/6$.
The function is odd and strictly increasing. Mathlib defines `Real.arcsin` on all of $\mathbb{R}$, constant outside $[-1, 1]$.

## Explanations

### L1 human

The arcsine of $x$ is the angle between $-\pi/2$ and $\pi/2$ whose sine is $x$.