Inverse trigonometric functions

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 $\arcsin \colon [-1, 1] \to [-\pi/2, \pi/2]$, the arccosine $\arccos \colon [-1, 1] \to [0, \pi]$ and the arctangent $\arctan \colon \mathbb{R} \to (-\pi/2, \pi/2)$.

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.