Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions

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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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
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

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/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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
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	$\sec$	$:$	$\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)$	$\to$	$\mathbb {R} \setminus (-1,1)=(-\infty ,-1]\cup [1,\infty )$	$\operatorname {arcsec}$	$:$	$\mathbb {R} \setminus (-1,1)$	$\to$	$[\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}$
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

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

The symbol $\,\setminus \,$ denotes set subtraction so that, for instance, $\mathbb {R} \setminus (-1,1)$ is the set of points in $\mathbb {R}$ (that is, real numbers) that are not in the interval $(-1,1),$ which in this case is equal to the set of all points in the two intervals $(-\infty ,-1]\cup [1,\infty ).$ Similarly, $[0,\pi ]\setminus \left\{{\tfrac {\pi }{2}}\right\}$ is the set of all points in the interval $[0,\pi ]$ that are not equal to ${\tfrac {\pi }{2}}$ while $\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}$ is the set of all points in $\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]$ that are not equal to $0,$ which can be written in terms of intervals as $$\left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}=\left[-{\tfrac {\pi }{2}},0\right)\cup \left(0,{\tfrac {\pi }{2}}\right]\qquad {\text{ and }}\qquad [0,\pi ]\setminus \left\{{\tfrac {\pi }{2}}\right\}=\left[0,{\tfrac {\pi }{2}}\,\right)\cup \left({\tfrac {\pi }{2}},\pi \right].$$

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 of all angles $\theta$ at which $\sin \theta \neq 0,$ which can also be written as $${\begin{alignedat}{4}\pi \mathbb {Z} +(0,\pi )&~=~\cdots \cup (-3\pi ,-2\pi )\cup (-2\pi ,-\pi )\cup (-\pi ,0)\cup (0,\pi )\cup (\pi ,2\pi )\cup (2\pi ,3\pi )\cup (3\pi ,4\pi )\cup \cdots \\&~=~\mathbb {R} \setminus \pi \mathbb {Z} .\\\end{alignedat}}$$ 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 of all angles $\theta$ at which $\cos \theta \neq 0,$ which can also be written as $${\begin{alignedat}{4}\pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)&~=~\cdots \cup \left(-{\tfrac {5\pi }{2}},-{\tfrac {3\pi }{2}}\right)\cup \left(-{\tfrac {3\pi }{2}},-{\tfrac {\pi }{2}}\right)\cup \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)\cup \left({\tfrac {\pi }{2}},{\tfrac {3\pi }{2}}\right)\cup \left({\tfrac {3\pi }{2}},{\tfrac {5\pi }{2}}\right)\cup \left({\tfrac {5\pi }{2}},{\tfrac {7\pi }{2}}\right)\cup \cdots \\&~=~\mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)\\\end{alignedat}}$$ where ${\textstyle \mathbb {R} \setminus \left({\tfrac {\pi }{2}}+\pi \mathbb {Z} \right)}$ is the set of all real numbers that do not belong to the set $${\begin{alignedat}{4}{\tfrac {\pi }{2}}+\pi \mathbb {Z} ~:&=~\left\{{\tfrac {\pi }{2}}+\pi n:n\in \mathbb {Z} \right\}\\[0.3ex]~&=~\left\{\ldots ,\,{\tfrac {\pi }{2}}-3\pi ,\,~~{\tfrac {\pi }{2}}-2\pi ,\,\,~\,~{\tfrac {\pi }{2}}-\pi ,\,~~{\tfrac {\pi }{2}},\,~\,~{\tfrac {\pi }{2}}+\pi ,\,~\,\,~\,{\tfrac {\pi }{2}}+2\pi ,\,~\,\,\,\,{\tfrac {\pi }{2}}+3\pi ,\,\ldots \right\}\\[0.6ex]~&=~\left\{\ldots ,~~\,~-{\tfrac {5\pi }{2}},~~\,~~\;-{\tfrac {3\pi }{2}},~\,~~\,~~-{\tfrac {\pi }{2}},\,~~{\tfrac {\pi }{2}},~~\,~~\,~~\;{\tfrac {3\pi }{2}},~~\,~~\,~~\;~~\,~{\tfrac {5\pi }{2}},~\,~~\,~~\,~~\,~~{\tfrac {7\pi }{2}},\,\ldots \right\}\\[0.6ex]~&=~\{\theta \in \mathbb {R} \;:\;\cos \theta =0\}.\\[0.3ex]\end{alignedat}}$$ In other words, the domain of $\,\tan \,$ and $\,\sec \,$ is the set of all real numbers that are not of the form ${\textstyle {\tfrac {\pi }{2}}+\pi n={\tfrac {\pi +2\pi n}{2}}={\tfrac {\pi (1+2n)}{2}}}$ for some integer $n;$ this is also the set of all numbers that are not of the form ${\textstyle {\tfrac {o}{2}}\pi }$ for some odd integer $o.$ These points not in the domain (meaning ${\textstyle {\tfrac {\pi }{2}}+\pi n}$ 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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