4.2: Graphing Inverse Trigonometric Functions

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 $y=x$.

Figure $\PageIndex{1}$: The sine function and inverse sine (or arcsine) function

Figure $\PageIndex{2}$: The cosine function and inverse cosine (or arccosine) function

Figure $\PageIndex{3}$: The tangent function and inverse tangent (or arctangent) function

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.