4.2: Graphing Inverse Trigonometric Functions
- Page ID
- 61258
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(y=x\).
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.