---
tag: 02HTDK1FBD
kind: definition
name: Inverse trigonometric functions
msc: [33B10]
defines: 05WEYX99X4
uses: [0B89D93YA1, 0RDP2HEFEW, 065XW2V6C3]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    locator: Principal values
---

## Definition

The inverse trigonometric functions are the inverses of the restrictions of the trigonometric functions. The restrictions are to the following intervals:
$\sin$ on $[-\pi/2, \pi/2]$, $\cos$ on $[0, \pi]$, and $\tan$ on $(-\pi/2, \pi/2)$.
They are the [[arcsine|0B89D93YA1]] $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$, the [[arccosine|0RDP2HEFEW]] $\arccos \colon [-1, 1] \to [0, \pi]$ and the [[arctangent|065XW2V6C3]] $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$.

## Remarks

The trigonometric functions are not injective on $\mathbb{R}$. The restrictions to these intervals are strictly monotone and continuous, so they have continuous inverses.
The values are the principal values. Other notations are $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$.
Similar definitions give $\operatorname{arccot}$, $\operatorname{arcsec}$ and $\operatorname{arccsc}$.

## Explanations

### L1 human

An inverse trigonometric function gives back an angle from a value of sine, cosine or tangent.
We restrict to one interval so that the answer is unique.