Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions

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{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions}}

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Original function	Description	Abbreviation		Domain		Image/range	Inverse function		Domain of inverse		Range of usual principal values of inverse
sine	${\frac {\mathbf {opp} \!\operatorname {osite} \quad \;}{\mathbf {hyp} \!\operatorname {otenuse} }}$	$\sin$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arcsin$	$:$	$[-1,1]$	$\to$	$\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]$
cosine	${\frac {\mathbf {adj} \!\operatorname {acent} }{\operatorname {hyp} \qquad \;}}$	$\cos$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arccos$	$:$	$[-1,1]$	$\to$	$[0,\pi ]$
tangent	${\frac {\operatorname {opp} }{\operatorname {adj} }}={\frac {\sin }{\cos }}$	$\tan$	$:$	$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$	$\to$	$\mathbb {R}$	$\arctan$	$:$	$\mathbb {R}$	$\to$	$\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$
cotangent	${\frac {\operatorname {adj} }{\operatorname {opp} }}={\frac {\cos }{\sin }}$	$\cot$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R}$	$\operatorname {arccot}$	$:$	$\mathbb {R}$	$\to$	$(0,\pi )$
secant	${\frac {\operatorname {hyp} }{\operatorname {adj} }}={\frac {1}{\cos }}$	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$	$\to$	$(-\infty ,-1]\cup [1,\infty )$	$\operatorname {arcsec}$	$:$	$(-\infty ,-1]\cup [1,\infty )$	$\to$	$\left[0,{\frac {\pi }{2}}\right)\cup \left({\frac {\pi }{2}},\pi \right]$
cosecant	${\frac {\operatorname {hyp} }{\operatorname {opp} }}={\frac {1}{\sin }}$	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$(-\infty ,-1]\cup [1,\infty )$	$\operatorname {arccsc}$	$:$	$(-\infty ,-1]\cup [1,\infty )$	$\to$	$\left[-{\frac {\pi }{2}},0\right)\cup \left(0,{\frac {\pi }{2}}\right]$

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{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions|includeTableDescription=true}}

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The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.{|class="wikitable" style="background-color: #FFFFFF; text-align: center; ;" |- ! style='border-style: solid none solid solid;'|Original
function ! style='border-style: solid none;'|Description ! style='border-style: solid none;'|Abbreviation ! style='border-style: solid none;' | ! style='border-style: solid none;' |Domain ! style='border-style: solid none;' | ! style='border-style: solid none;' |Image/range ! style='border-style: solid none solid solid;'|Inverse
function ! style='border-style: solid none;' | ! style='border-style: solid none;' |Domain of
inverse ! style='border-style: solid none;' | ! style='border-style: solid none;' |Range of usual
principal values of inverse |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|sine | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\mathbf {opp} \!\operatorname {osite} \quad \;}{\mathbf {hyp} \!\operatorname {otenuse} }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\sin$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[-1,1]$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arcsin$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$[-1,1]$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cosine | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\mathbf {adj} \!\operatorname {acent} }{\operatorname {hyp} \qquad \;}}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\cos$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[-1,1]$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arccos$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$[-1,1]$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[0,\pi ]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|tangent | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {opp} }{\operatorname {adj} }}={\frac {\sin }{\cos }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\tan$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\mathbb {R}$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arctan$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cotangent | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {adj} }{\operatorname {opp} }}={\frac {\cos }{\sin }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\cot$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +(0,\pi )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\mathbb {R}$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arccot}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(0,\pi )$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|secant | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {hyp} }{\operatorname {adj} }}={\frac {1}{\cos }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\sec$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arcsec}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[0,{\frac {\pi }{2}}\right)\cup \left({\frac {\pi }{2}},\pi \right]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cosecant | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {hyp} }{\operatorname {opp} }}={\frac {1}{\sin }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\csc$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +(0,\pi )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arccsc}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[-{\frac {\pi }{2}},0\right)\cup \left(0,{\frac {\pi }{2}}\right]$ |}

With includeTableDescription and includeExplanationOfNotation

Calling

{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions|includeTableDescription=true|includeExplanationOfNotation=true}}

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The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.{|class="wikitable" style="background-color: #FFFFFF; text-align: center; ;" |- ! style='border-style: solid none solid solid;'|Original
function ! style='border-style: solid none;'|Description ! style='border-style: solid none;'|Abbreviation ! style='border-style: solid none;' | ! style='border-style: solid none;' |Domain ! style='border-style: solid none;' | ! style='border-style: solid none;' |Image/range ! style='border-style: solid none solid solid;'|Inverse
function ! style='border-style: solid none;' | ! style='border-style: solid none;' |Domain of
inverse ! style='border-style: solid none;' | ! style='border-style: solid none;' |Range of usual
principal values of inverse |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|sine | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\mathbf {opp} \!\operatorname {osite} \quad \;}{\mathbf {hyp} \!\operatorname {otenuse} }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\sin$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[-1,1]$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arcsin$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$[-1,1]$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cosine | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\mathbf {adj} \!\operatorname {acent} }{\operatorname {hyp} \qquad \;}}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\cos$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[-1,1]$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arccos$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$[-1,1]$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$[0,\pi ]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|tangent | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {opp} }{\operatorname {adj} }}={\frac {\sin }{\cos }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\tan$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\mathbb {R}$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\arctan$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cotangent | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {adj} }{\operatorname {opp} }}={\frac {\cos }{\sin }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\cot$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +(0,\pi )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\mathbb {R}$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arccot}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$\mathbb {R}$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(0,\pi )$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|secant | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {hyp} }{\operatorname {adj} }}={\frac {1}{\cos }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\sec$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arcsec}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[0,{\frac {\pi }{2}}\right)\cup \left({\frac {\pi }{2}},\pi \right]$ |- | style='border-style: solid none solid solid; text-align: left; padding-left: 0.5em;'|cosecant | style='border-style: solid none; text-align: left; padding-left: 0.5em;'|${\frac {\operatorname {hyp} }{\operatorname {opp} }}={\frac {1}{\sin }}$ | style='border-style: solid none; text-align: right; padding-left: 2em;'|$\csc$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: left;' |$\pi \mathbb {Z} +(0,\pi )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none solid solid; text-align: right; padding-left: 2em;'|$\operatorname {arccsc}$ | style='border-style: solid none; text-align: right;' |$:$ | style='border-style: solid none; text-align: center;'|$(-\infty ,-1]\cup [1,\infty )$ | style='border-style: solid none; text-align: right;' |$\to$ | style='border-style: solid none; text-align: left;' |$\left[-{\frac {\pi }{2}},0\right)\cup \left(0,{\frac {\pi }{2}}\right]$ |} The symbol $\mathbb {R} =(-\infty ,\infty )$ denotes the set of all real numbers and $\mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}$ denotes the set of all integers. The set of all integer multiples of $\pi$ is denoted by $$\pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,3\pi ,\,4\pi ,\,\ldots \}~=~\{\theta \in \mathbb {R} \;:\;\sin \theta =0\}.$$ The Minkowski sum notation $\pi \mathbb {Z} +(0,\pi )$ means $${\begin{alignedat}{4}\pi \mathbb {Z} +(0,\pi )&~=~\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup (2\pi ,3\pi )\cup (3\pi ,4\pi )\cup \cdots \\&~=~\mathbb {R} \setminus \pi \mathbb {Z} \\\end{alignedat}}$$ where $\,\setminus \,$ denotes set subtraction. In other words, the domain of $\,\cot \,$ and $\,\csc \,$ is the set $\mathbb {R} \setminus \pi \mathbb {Z}$ of all real numbers that are not of the form $\pi n$ for some integer $n.$

Similarly, the domain of $\,\tan \,$ and $\,\sec \,$ is the set $${\begin{alignedat}{4}\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)&~=~\cdots \cup \left(-{\frac {3\pi }{2}},-{\frac {\pi }{2}}\right)\cup \left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)\cup \left({\frac {\pi }{2}},{\frac {3\pi }{2}}\right)\cup \left({\frac {3\pi }{2}},{\frac {5\pi }{2}}\right)\cup \left({\frac {5\pi }{2}},{\frac {7\pi }{2}}\right)\cup \cdots \\&~=~\mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{alignedat}}$$ where $\mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)$ is the set of all real numbers that do not belong to the set $${\frac {\pi }{2}}+\pi \mathbb {Z} ~:=~\left\{{\frac {\pi }{2}}+\pi n:n\in \mathbb {Z} \right\}=\left\{\ldots ,\,-{\frac {3\pi }{2}},\,-{\frac {\pi }{2}},\,{\frac {\pi }{2}},\,{\frac {3\pi }{2}},\,{\frac {5\pi }{2}},\,{\frac {7\pi }{2}},\,\ldots \right\}~=~\{\theta \in \mathbb {R} \;:\;\cos \theta =0\};$$ said differently, the domain of $\,\tan \,$ and $\,\sec \,$ is the set of all real numbers that are not of the form ${\frac {\pi }{2}}+\pi n$ for some integer $n.$

See also

• Template:Trigonometric Functions Exact Values Table

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