Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions

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{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions}}

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Name	Symbol		Domain		Image	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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[0,\pi \right]\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \left\{0\right\}$

With includeTableDescription

Calling

{{DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions|includeTableDescription=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	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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[0,\pi \right]\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \left\{0\right\}$

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	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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[0,\pi \right]\setminus \left\{{\tfrac {\pi }{2}}\right\}$
cosecant	$\csc$	$:$	$\pi \mathbb {Z} +(0,\pi )$	$\to$	$\mathbb {R} \setminus (-1,1)$	$\operatorname {arccsc}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \left\{0\right\}$

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

$$\pi \mathbb {Z} ~:=~\{\pi n\;:\;n\in \mathbb {Z} \}~=~\{\ldots ,\,-3\pi \,-2\pi ,\,-\pi ,\,0,\,\pi ,\,2\pi ,\,3\pi ,\,4\pi ,\,\ldots \}~=~\{\theta \in \mathbb {R} \;:\;\sin \theta =0\}.$$

The Minkowski sum notation ${\textstyle \pi \mathbb {Z} +(0,\pi )}$ and $\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)$ 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 $${\begin{alignedat}{4}\pi \mathbb {Z} +(0,\pi )&~=~\cdots \cup (-3\pi ,-2\pi )\cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup (2\pi ,3\pi )\cup (3\pi ,4\pi )\cup \cdots \\&~=~\mathbb {R} \setminus \pi \mathbb {Z} \\\end{alignedat}}$$

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 $n.$ These points not in the domain (meaning $\pi n$ for $n$ an integer) are exactly those numbers $\theta$ at which $0=\sin \theta ;$ this is because these are also exactly the $\theta$ at which ${\textstyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}}$ and ${\textstyle \csc \theta ={\frac {1}{\sin \theta }}}$ would be divided by $0.$

Domain of tangent $\tan$ and secant $\sec$: The domains of $\,\tan \,$ and $\,\sec \,$ are the same. They are the set $${\begin{alignedat}{4}\pi \mathbb {Z} +\left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)&~=~\cdots \cup \left(-{\frac {5\pi }{2}},-{\frac {3\pi }{2}}\right)\cup \left(-{\frac {3\pi }{2}},-{\frac {\pi }{2}}\right)\cup \left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)\cup \left({\frac {\pi }{2}},{\frac {3\pi }{2}}\right)\cup \left({\frac {3\pi }{2}},{\frac {5\pi }{2}}\right)\cup \left({\frac {5\pi }{2}},{\frac {7\pi }{2}}\right)\cup \cdots \\&~=~\mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{alignedat}}$$

where ${\textstyle \mathbb {R} \setminus \left({\frac {\pi }{2}}+\pi \mathbb {Z} \right)}$ is the set of all real numbers that do not belong to the set $${\begin{alignedat}{4}{\frac {\pi }{2}}+\pi \mathbb {Z} ~:&=~\left\{{\frac {\pi }{2}}+\pi n:n\in \mathbb {Z} \right\}\\[0.3ex]~&=~\left\{\ldots ,\,{\frac {\pi }{2}}-3\pi ,\,~~{\frac {\pi }{2}}-2\pi ,\,\,~\,~{\frac {\pi }{2}}-\pi ,\,~~{\frac {\pi }{2}},\,~\,~{\frac {\pi }{2}}+\pi ,\,~\,\,~\,{\frac {\pi }{2}}+2\pi ,\,~\,\,\,\,{\frac {\pi }{2}}+3\pi ,\,\ldots \right\}\\[0.6ex]~&=~\left\{\ldots ,~~\,~-{\frac {5\pi }{2}},~~\,~~-{\frac {3\pi }{2}},~\,~~\,~~-{\frac {\pi }{2}},\,~~{\frac {\pi }{2}},~~\,~~\,~~\;{\frac {3\pi }{2}},~~\,~~\,~~\;~~\,~{\frac {5\pi }{2}},~\,~~\,~~\,~~\,~~{\frac {7\pi }{2}},\,\ldots \right\}\\[0.6ex]~&=~\{\theta \in \mathbb {R} \;:\;\cos \theta =0\}\\[0.3ex]\end{alignedat}}$$

said differently, the domain of $\,\tan \,$ and $\,\sec \,$ is the set of all real numbers that are not of the form ${\textstyle {\frac {\pi }{2}}+\pi n={\frac {\pi +2\pi n}{2}}={\frac {\pi (1+2n)}{2}}}$ for some integer $n;$ this is also the set of all numbers that are not of the form ${\textstyle {\frac {o}{2}}\pi }$ for some odd integer $o.$ These points not in the domain (meaning ${\textstyle {\frac {\pi }{2}}+\pi n}$ for $n$ an integer) are exactly those numbers $\theta$ at which $0=\cos \theta ;$ this is because these are also exactly the $\theta$ at which ${\textstyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}$ and ${\textstyle \sec \theta ={\frac {1}{\cos \theta }}}$ would be divided by $0.$

See also

• Template:Trigonometric Functions Exact Values Table

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