The inverse trigonometric functions are connected to the unit circle definitions of the trigonometric functions and their mapping diagrams.

TRIG.INV.UC: Inverse Connections to The Unit Circle.

As with all inverse functions , the inverse trigonometric functions are connected visually to the mapping diagrams of the core trigonometric functions, sine , cosine, and tangent.

(i) $\sin^{-1}$ or $\arcsin$

(ii) $\cos^{-1}$ or $\arccos$

(iii) $\tan^{-1}$ or $\arctan$

(i) $\sin^{-1}$ or $\arcsin$
Suppose $x \in [-1,1]$.
Definition of arcsine function:
The $arcsine$ (or inverse sine, $sin^{-1}$) function, $\arcsin : [-1,1] \rightarrow [-\frac {\pi}2, \frac {\pi}2]$ , is defined as the inverse function for the sine function for real numbers, $t$, $-\frac {\pi}2 \le t \le \frac{\pi}2$
$\arcsin(x) = t$ if and only if $\sin(t) = x$ and $-\frac {\pi}2 \le t \le \frac{\pi}2$.

(ii) $\cos^{-1}$ or $\arccos$
Suppose $x \in [-1,1]$.
Definition of arccosine function: The $arccosine$ (or inverse cosine, $cos^{-1}$) function, $\arccos : [-1,1] \rightarrow [0,\pi]$ , is defined as the inverse function for the cosine function for real numbers, $t$, $0 \le t \le \pi$
$\arccos(x) = t$ if and only if $\cos(t) = x$ and $0 \le t \le \pi$

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(iii) $\tan^{-1}$ or $\arctan$
Suppose $x \in (-\infty, \infty)$.

Definition of arctangent function:
The $arctangent$ (or inverse tangent, $\tan^{-1}$) function, $\arctan : (-\infty, \infty) \rightarrow R$ , is defined as the inverse function for the > for real numbers, $t$, $-\frac {\pi}2 < t < \frac{\pi}2$
$\arctan(x) = t$ if and only if $\tan(t) = x$ and $-\frac {\pi}2 < t < \frac{\pi}2$.

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GeoGebra Table, Graph and Mapping Diagram for arctangent, arccosine, and arcsine.
Use the slider to choose
Example=1: arcsine        Example=2: arccosine      Example=3: arctangent