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 $
.
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 $.
.
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