Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions/doc

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{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions}}

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Name	Symbol		Domain		Image/Range	Inverse function		Domain		Image of principal values
sine	$\sin$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arcsin$	$:$	$[-1,1]$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]$
cosine	$\cos$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arccos$	$:$	$[-1,1]$	$\to$	$[0,\pi ]$
tangent	$\tan$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R}$	$\arctan$	$:$	$\mathbb {R}$	$\to$	$\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$
cotangent	$\cot$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R}$	$\operatorname {arccot}$	$:$	$\mathbb {R}$	$\to$	$(0,\pi )$
secant	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}$

With includeTableDescription

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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.

Name	Symbol		Domain		Image/Range	Inverse function		Domain		Image of principal values
sine	$\sin$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arcsin$	$:$	$[-1,1]$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]$
cosine	$\cos$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arccos$	$:$	$[-1,1]$	$\to$	$[0,\pi ]$
tangent	$\tan$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R}$	$\arctan$	$:$	$\mathbb {R}$	$\to$	$\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$
cotangent	$\cot$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R}$	$\operatorname {arccot}$	$:$	$\mathbb {R}$	$\to$	$(0,\pi )$
secant	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}$

With includeTableDescription and includeExplanationOfNotation

Calling

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

will display:

The table below displays names and domains of the inverse trigonometric functions along with the range of their usual principal values in radians.

Name	Symbol		Domain		Image/Range	Inverse function		Domain		Image of principal values
sine	$\sin$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arcsin$	$:$	$[-1,1]$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]$
cosine	$\cos$	$:$	$\mathbb {R}$	$\to$	$[-1,1]$	$\arccos$	$:$	$[-1,1]$	$\to$	$[0,\pi ]$
tangent	$\tan$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R}$	$\arctan$	$:$	$\mathbb {R}$	$\to$	$\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$
cotangent	$\cot$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R}$	$\operatorname {arccot}$	$:$	$\mathbb {R}$	$\to$	$(0,\pi )$
secant	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}$

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

$$\pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,\ldots \}.$$

The symbol $\,\setminus \,$ denotes set subtraction so that, for instance, $\mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )$ is the set of points in $\mathbb {R}$ (that is, real numbers) that are not in the interval $(-1,1).$

The Minkowski sum notation ${\textstyle \pi \mathbb {Z} +(0,\pi )}$ and $\pi \mathbb {Z} +{\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}$ that is used above to concisely write the domains of $\cot ,\csc ,\tan ,{\text{ and }}\sec$ is now explained.

Domain of cotangent $\cot$ and cosecant $\csc$: The domains of $\,\cot \,$ and $\,\csc \,$ are the same. They are the set of all angles $\theta$ at which $\sin \theta \neq 0,$ i.e. all real numbers that are not of the form $\pi n$ for some integer $n,$

$${\begin{aligned}\pi \mathbb {Z} +(0,\pi )&=\cdots \cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup \cdots \\&=\mathbb {R} \setminus \pi \mathbb {Z} \end{aligned}}$$

Domain of tangent $\tan$ and secant $\sec$: The domains of $\,\tan \,$ and $\,\sec \,$ are the same. They are the set of all angles $\theta$ at which $\cos \theta \neq 0,$

$${\begin{aligned}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&=\cdots \cup {\bigl (}{-{\tfrac {3\pi }{2}}},{-{\tfrac {\pi }{2}}}{\bigr )}\cup {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )}\cup {\bigl (}{\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}{\bigr )}\cup \cdots \\&=\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{aligned}}$$

See also

• Template:Trigonometric Functions Exact Values Table