4.2: Graphing Inverse Trigonometric Functions

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Learning Objectives

Understand the meaning of restricted domain as it applies to the inverses of the six trigonometric functions.
Apply the domain, range, and quadrants of the six inverse trigonometric functions to evaluate expressions.

we can define the inverse trigonometric functions.

The inverse sine function $y={\sin}^{−1}x$ means $x=\sin\space y$. The inverse sine function is sometimes called the arcsine function, and notated $\arcsin\space x$.
$y={\sin}^{−1}x$ has domain $[−1,1]$ and range $\left[−\frac{\pi}{2},\frac{\pi}{2}\right]$
The inverse cosine function $y={\cos}^{−1}x$ means $x=\cos\space y$. The inverse cosine function is sometimes called the arccosine function, and notated $\arccos\space x$.
$y={\cos}^{−1}x$ has domain $[−1,1]$ and range $[0,π]$
The inverse tangent function $y={\tan}^{−1}x$ means $x=\tan\space y$. The inverse tangent function is sometimes called the arctangent function, and notated $\arctan\space x$.
$y={\tan}^{−1}x$ has domain $(−\infty,\infty)$ and range $\left(−\frac{\pi}{2},\frac{\pi}{2}\right)$

The graphs of the inverse functions are shown in Figures $\PageIndex{1}$ - $\PageIndex{3}$. Notice that the output of each of these inverse functions is a number, an angle in radian measure. We see that ${\sin}^{−1}x$ has domain $[ −1,1 ]$ and range $\left[ −\dfrac{\pi}{2},\dfrac{\pi}{2} \right]$, ${\cos}^{−1}x$ has domain $[ −1,1 ]$ and range $[0,\pi]$, and ${\tan}^{−1}x$ has domain of all real numbers and range $\left(−\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$. To find the domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph of the inverse trigonometric function is a reflection of the graph of the original function about the line $y=x$.

A graph of the functions of sine of x and arc sine of x. There is a dotted line y=x between the two graphs, to show inverse nature of the two functions — Figure $\PageIndex{1}$: The sine function and inverse sine (or arcsine) function

A graph of the functions of cosine of x and arc cosine of x. There is a dotted line at y=x to show the inverse nature of the two functions. — Figure $\PageIndex{2}$: The cosine function and inverse cosine (or arccosine) function

A graph of the functions of tangent of x and arc tangent of x. There is a dotted line at y=x to show the inverse nature of the two functions. — Figure $\PageIndex{3}$: The tangent function and inverse tangent (or arctangent) function

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RELATIONS FOR INVERSE SINE, COSINE, AND TANGENT FUNCTIONS

For angles in the interval $\left[ −\dfrac{\pi}{2},\dfrac{\pi}{2} \right]$, if $\sin y=x$, then ${\sin}^{−1}x=y$.

For angles in the interval $[ 0,\pi ]$, if $\cos y=x$, then ${\cos}^{−1}x=y$.

For angles in the interval $\left(−\dfrac{\pi}{2},\dfrac{\pi}{2}\right )$, if $\tan y=x$,then ${\tan}^{−1}x=y$.

RELATIONS FOR INVERSE SINE, COSINE, AND TANGENT FUNCTIONS

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.

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