Inverse trigonometric functions

Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore, the result ranges of the inverse functions are proper (i.e. strict) subsets of the domains of the original functions.

For example, using function in the sense of multivalued functions, just as the square root function ${\displaystyle y={\sqrt {x}}}$ could be defined from ${\displaystyle y^{2}=x,}$ the function ${\displaystyle y=\arcsin(x)}$ is defined so that ${\displaystyle \sin(y)=x.}$ For a given real number ${\displaystyle x,}$ with ${\displaystyle -1\leq x\leq 1,}$ there are multiple (in fact, countably infinitely many) numbers ${\displaystyle y}$ such that ${\displaystyle \sin(y)=x}$; for example, ${\displaystyle \sin(0)=0,}$ but also ${\displaystyle \sin(\pi )=0,}$ ${\displaystyle \sin(2\pi )=0,}$ etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each ${\displaystyle x}$ in the domain, the expression ${\displaystyle \arcsin(x)}$ will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

Name	Usual notation	Definition	Domain of ${\displaystyle x}$ for real result	Range of usual principal value (radians)	Range of usual principal value (degrees)
arcsine	${\displaystyle y=\arcsin(x)}$	x = sin(y)	${\displaystyle -1\leq x\leq 1}$	${\displaystyle -{\frac {\pi }{2}}\leq y\leq {\frac {\pi }{2}}}$	${\displaystyle -90^{\circ }\leq y\leq 90^{\circ }}$
arccosine	${\displaystyle y=\arccos(x)}$	x = cos(y)	${\displaystyle -1\leq x\leq 1}$	${\displaystyle 0\leq y\leq \pi }$	${\displaystyle 0^{\circ }\leq y\leq 180^{\circ }}$
arctangent	${\displaystyle y=\arctan(x)}$	x = tan(y)	all real numbers	${\displaystyle -{\frac {\pi }{2}}<y<{\frac {\pi }{2}}}$	${\displaystyle -90^{\circ }<y<90^{\circ }}$
arccotangent	${\displaystyle y=\operatorname {arccot}(x)}$	x = cot(y)	all real numbers	${\displaystyle 0<y<\pi }$	${\displaystyle 0^{\circ }<y<180^{\circ }}$
arcsecant	${\displaystyle y=\operatorname {arcsec}(x)}$	x = sec(y)	${\displaystyle {\left\vert x\right\vert }\geq 1}$	${\displaystyle 0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi }$	${\displaystyle 0^{\circ }\leq y<90^{\circ }{\text{ or }}90^{\circ }<y\leq 180^{\circ }}$
arccosecant	${\displaystyle y=\operatorname {arccsc}(x)}$	x = csc(y)	${\displaystyle {\left\vert x\right\vert }\geq 1}$	${\displaystyle -{\frac {\pi }{2}}\leq y<0{\text{ or }}0<y\leq {\frac {\pi }{2}}}$	${\displaystyle -90^{\circ }\leq y<0^{\circ }{\text{ or }}0^{\circ }<y\leq 90^{\circ }}$

Note: Some authors define the range of arcsecant to be ${\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}\pi \leq y<{\frac {3\pi }{2}})}$, because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, ${\displaystyle \tan(\operatorname {arcsec}(x))={\sqrt {x^{2}-1}},}$ whereas with the range ${\textstyle (0\leq y<{\frac {\pi }{2}}{\text{ or }}{\frac {\pi }{2}}<y\leq \pi )}$, we would have to write ${\displaystyle \tan(\operatorname {arcsec}(x))=\pm {\sqrt {x^{2}-1}},}$ since tangent is nonnegative on ${\textstyle 0\leq y<{\frac {\pi }{2}},}$ but nonpositive on ${\textstyle {\frac {\pi }{2}}<y\leq \pi .}$ For a similar reason, the same authors define the range of arccosecant to be ${\textstyle (-\pi <y\leq -{\frac {\pi }{2}}}$ or ${\textstyle 0<y\leq {\frac {\pi }{2}}).}$

If ${\displaystyle x}$ is allowed to be a complex number, then the range of ${\displaystyle y}$ applies only to its real part.

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	${\displaystyle \sin }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} }$	${\displaystyle \to }$	${\displaystyle [-1,1]}$	${\displaystyle \arcsin }$	${\displaystyle$ :}	${\displaystyle [-1,1]}$	${\displaystyle \to }$	${\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]}$
cosine	${\displaystyle \cos }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} }$	${\displaystyle \to }$	${\displaystyle [-1,1]}$	${\displaystyle \arccos }$	${\displaystyle$ :}	${\displaystyle [-1,1]}$	${\displaystyle \to }$	${\displaystyle [0,\pi ]}$
tangent	${\displaystyle \tan }$	${\displaystyle$ :}	${\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}$	${\displaystyle \to }$	${\displaystyle \mathbb {R} }$	${\displaystyle \arctan }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} }$	${\displaystyle \to }$	${\displaystyle \left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}$
cotangent	${\displaystyle \cot }$	${\displaystyle$ :}	${\displaystyle \pi \mathbb {Z} +(0,\pi )}$	${\displaystyle \to }$	${\displaystyle \mathbb {R} }$	${\displaystyle \operatorname {arccot} }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} }$	${\displaystyle \to }$	${\displaystyle (0,\pi )}$
secant	${\displaystyle \sec }$	${\displaystyle$ :}	${\displaystyle \pi \mathbb {Z} +\left(-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right)}$	${\displaystyle \to }$	${\displaystyle \mathbb {R} \setminus (-1,1)}$	${\displaystyle \operatorname {arcsec} }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} \setminus (-1,1)}$	${\displaystyle \to }$	${\displaystyle [\,0,\;\pi \,]\;\;\;\setminus \left\{{\tfrac {\pi }{2}}\right\}}$
cosecant	${\displaystyle \csc }$	${\displaystyle$ :}	${\displaystyle \pi \mathbb {Z} +(0,\pi )}$	${\displaystyle \to }$	${\displaystyle \mathbb {R} \setminus (-1,1)}$	${\displaystyle \operatorname {arccsc} }$	${\displaystyle$ :}	${\displaystyle \mathbb {R} \setminus (-1,1)}$	${\displaystyle \to }$	${\displaystyle \left[-{\tfrac {\pi }{2}},{\tfrac {\pi }{2}}\right]\setminus \{0\}}$

The symbol ${\displaystyle \mathbb {R} =(-\infty ,\infty )}$ denotes the set of all real numbers and ${\displaystyle \mathbb {Z} =\{\ldots ,\,-2,\,-1,\,0,\,1,\,2,\,\ldots \}}$ denotes the set of all integers. The set of all integer multiples of ${\displaystyle \pi }$ is denoted by

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

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

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

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

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

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

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

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of ${\displaystyle 2\pi$ :}

• Sine and cosecant begin their period at ${\textstyle 2\pi k-{\frac {\pi }{2}}}$ (where ${\displaystyle k}$ is an integer), finish it at ${\textstyle 2\pi k+{\frac {\pi }{2}},}$ and then reverse themselves over ${\textstyle 2\pi k+{\frac {\pi }{2}}}$ to ${\textstyle 2\pi k+{\frac {3\pi }{2}}.}$
• Cosine and secant begin their period at ${\displaystyle 2\pi k,}$ finish it at ${\displaystyle 2\pi k+\pi .}$ and then reverse themselves over ${\displaystyle 2\pi k+\pi }$ to ${\displaystyle 2\pi k+2\pi .}$
• Tangent begins its period at ${\textstyle 2\pi k-{\frac {\pi }{2}},}$ finishes it at ${\textstyle 2\pi k+{\frac {\pi }{2}},}$ and then repeats it (forward) over ${\textstyle 2\pi k+{\frac {\pi }{2}}}$ to ${\textstyle 2\pi k+{\frac {3\pi }{2}}.}$
• Cotangent begins its period at ${\displaystyle 2\pi k,}$ finishes it at ${\displaystyle 2\pi k+\pi ,}$ and then repeats it (forward) over ${\displaystyle 2\pi k+\pi }$ to ${\displaystyle 2\pi k+2\pi .}$

This periodicity is reflected in the general inverses, where ${\displaystyle k}$ is some integer.

The following table shows how inverse trigonometric functions may be used to solve equalities involving the six standard trigonometric functions. It is assumed that the given values ${\displaystyle \theta ,}$ ${\displaystyle r,}$ ${\displaystyle s,}$ ${\displaystyle x,}$ and ${\displaystyle y}$ all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some ${\displaystyle k\in \mathbb {Z} }$" is just another way of saying "for some integer ${\displaystyle k.}$"

The symbol ${\displaystyle \,\iff \,}$ is logical equality. The expression "LHS ${\displaystyle \,\iff \,}$ RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS ${\displaystyle \,\iff \,}$ RHS" would not have been written (see this footnote[note 1] for an example illustrating this concept).

Equation	if and only if	Solution
${\displaystyle \sin \theta =y}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$	${\displaystyle (-1)^{k}}$	${\displaystyle \arcsin(y)}$	${\displaystyle +}$		${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \csc \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$	${\displaystyle (-1)^{k}}$	${\displaystyle \operatorname {arccsc}(r)}$	${\displaystyle +}$		${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \cos \theta =x}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$	${\displaystyle \pm \,}$	${\displaystyle \arccos(x)}$	${\displaystyle +}$	${\displaystyle 2}$	${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \sec \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$	${\displaystyle \pm \,}$	${\displaystyle \operatorname {arcsec}(r)}$	${\displaystyle +}$	${\displaystyle 2}$	${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \tan \theta =s}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$		${\displaystyle \arctan(s)}$	${\displaystyle +}$		${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \cot \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =\,}$		${\displaystyle \operatorname {arccot}(r)}$	${\displaystyle +}$		${\displaystyle \pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$

where the first four solutions can be written in expanded form as:

Equation	if and only if	Solution
${\displaystyle \sin \theta =y}$	${\displaystyle \iff }$	${\displaystyle \theta =\;\;\;\,\arcsin(y)+2\pi h}$           or ${\displaystyle \theta =-\arcsin(y)+2\pi h+\pi }$	for some ${\displaystyle h\in \mathbb {Z} }$
${\displaystyle \csc \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =\;\;\;\,\operatorname {arccsc}(y)+2\pi h}$           or ${\displaystyle \theta =-\operatorname {arccsc}(y)+2\pi h+\pi }$	for some ${\displaystyle h\in \mathbb {Z} }$
${\displaystyle \cos \theta =x}$	${\displaystyle \iff }$	${\displaystyle \theta =\;\;\;\,\arccos(y)+2\pi k}$          or ${\displaystyle \theta =-\arccos(y)+2\pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$
${\displaystyle \sec \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =\;\;\;\,\operatorname {arcsec}(y)+2\pi k}$          or ${\displaystyle \theta =-\operatorname {arcsec}(y)+2\pi k}$	for some ${\displaystyle k\in \mathbb {Z} }$

For example, if ${\displaystyle \cos \theta =-1}$ then ${\displaystyle \theta =\pi +2\pi k=-\pi +2\pi (1+k)}$ for some ${\displaystyle k\in \mathbb {Z} .}$ While if ${\displaystyle \sin \theta =\pm 1}$ then ${\textstyle \theta ={\frac {\pi }{2}}+\pi k=-{\frac {\pi }{2}}+\pi (k+1)}$ for some ${\displaystyle k\in \mathbb {Z} ,}$ where ${\displaystyle k}$ will be even if ${\displaystyle \sin \theta =1}$ and it will be odd if ${\displaystyle \sin \theta =-1.}$ The equations ${\displaystyle \sec \theta =-1}$ and ${\displaystyle \csc \theta =\pm 1}$ have the same solutions as ${\displaystyle \cos \theta =-1}$ and ${\displaystyle \sin \theta =\pm 1,}$ respectively. In all equations above except for those just solved (i.e. except for ${\displaystyle \sin }$/${\displaystyle \csc \theta =\pm 1}$ and ${\displaystyle \cos }$/${\displaystyle \sec \theta =-1}$), the integer ${\displaystyle k}$ in the solution's formula is uniquely determined by ${\displaystyle \theta }$ (for fixed ${\displaystyle r,s,x,}$ and ${\displaystyle y}$).

The solutions to ${\displaystyle \cos \theta =x}$ and ${\displaystyle \sec \theta =x}$ involve the "plus or minus" symbol ${\displaystyle \,\pm ,\,}$ whose meaning is now clarified. Only the solution to ${\displaystyle \cos \theta =x}$ will be discussed since the discussion for ${\displaystyle \sec \theta =x}$ is the same. We are given ${\displaystyle x}$ between ${\displaystyle -1\leq x\leq 1}$ and we know that there is an angle ${\displaystyle \theta }$ in some interval that satisfies ${\displaystyle \cos \theta =x.}$ We want to find this ${\displaystyle \theta .}$ The table above indicates that the solution is

$${\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }$$

which is a shorthand way of saying that (at least) one of the following statement is true:

1. ${\displaystyle \,\theta =\arccos x+2\pi k\,}$ for some integer ${\displaystyle k,}$
   or
2. ${\displaystyle \,\theta =-\arccos x+2\pi k\,}$ for some integer ${\displaystyle k.}$

As mentioned above, if ${\displaystyle \,\arccos x=\pi \,}$ (which by definition only happens when ${\displaystyle x=\cos \pi =-1}$) then both statements (1) and (2) hold, although with different values for the integer ${\displaystyle k}$: if ${\displaystyle K}$ is the integer from statement (1), meaning that ${\displaystyle \theta =\pi +2\pi K}$ holds, then the integer ${\displaystyle k}$ for statement (2) is ${\displaystyle K+1}$ (because ${\displaystyle \theta =-\pi +2\pi (1+K)}$). However, if ${\displaystyle x\neq -1}$ then the integer ${\displaystyle k}$ is unique and completely determined by ${\displaystyle \theta .}$ If ${\displaystyle \,\arccos x=0\,}$ (which by definition only happens when ${\displaystyle x=\cos 0=1}$) then ${\displaystyle \,\pm \arccos x=0\,}$ (because ${\displaystyle \,+\arccos x=+0=0\,}$ and ${\displaystyle \,-\arccos x=-0=0\,}$ so in both cases ${\displaystyle \,\pm \arccos x\,}$ is equal to ${\displaystyle 0}$) and so the statements (1) and (2) happen to be identical in this particular case (and so both hold). Having considered the cases ${\displaystyle \,\arccos x=0\,}$ and ${\displaystyle \,\arccos x=\pi ,\,}$ we now focus on the case where ${\displaystyle \,\arccos x\neq 0\,}$ and ${\displaystyle \,\arccos x\neq \pi ,\,}$ So assume this from now on. The solution to ${\displaystyle \cos \theta =x}$ is still

$${\displaystyle \,\theta =\pm \arccos x+2\pi k\,\quad {\text{ for some }}k\in \mathbb {Z} }$$

which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because ${\displaystyle \,\arccos x\neq 0\,}$ and ${\displaystyle \,0<\arccos x<\pi ,\,}$ statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about ${\displaystyle \theta }$ is needed to determine which one holds. For example, suppose that ${\displaystyle x=0}$ and that all that is known about ${\displaystyle \theta }$ is that ${\displaystyle \,-\pi \leq \theta \leq \pi \,}$ (and nothing more is known). Then

$${\displaystyle \arccos x=\arccos 0={\frac {\pi }{2}}}$$

and moreover, in this particular case ${\displaystyle k=0}$ (for both the ${\displaystyle \,+\,}$ case and the ${\displaystyle \,-\,}$ case) and so consequently,

$${\displaystyle \theta ~=~\pm \arccos x+2\pi k~=~\pm \left({\frac {\pi }{2}}\right)+2\pi (0)~=~\pm {\frac {\pi }{2}}.}$$

This means that ${\displaystyle \theta }$ could be either ${\displaystyle \,\pi /2\,}$ or ${\displaystyle \,-\pi /2.}$ Without additional information it is not possible to determine which of these values ${\displaystyle \theta }$ has. An example of some additional information that could determine the value of ${\displaystyle \theta }$ would be knowing that the angle is above the ${\displaystyle x}$-axis (in which case ${\displaystyle \theta =\pi /2}$) or alternatively, knowing that it is below the ${\displaystyle x}$-axis (in which case ${\displaystyle \theta =-\pi /2}$).

The equations above can be transformed by using the reflection and shift identities:[18]

Transforming equations by shifts and reflections

Argument: ${\displaystyle {\underline {\;~~~~~~\;}}=}$	${\displaystyle -\theta }$	${\displaystyle {\frac {\pi }{2}}\pm \theta }$	${\displaystyle \pi \pm \theta }$	${\displaystyle {\frac {3\pi }{2}}\pm \theta }$	${\displaystyle 2k\pi \pm \theta ,}$ ${\displaystyle (k\in \mathbb {Z} )}$
${\displaystyle \sin {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle -\sin \theta }$	${\displaystyle {\phantom {-}}\cos \theta }$	${\displaystyle \mp \sin \theta }$	${\displaystyle -\cos \theta }$	${\displaystyle \pm \sin \theta }$
${\displaystyle \csc {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle -\csc \theta }$	${\displaystyle {\phantom {-}}\sec \theta }$	${\displaystyle \mp \csc \theta }$	${\displaystyle -\sec \theta }$	${\displaystyle \pm \csc \theta }$
${\displaystyle \cos {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle {\phantom {-}}\cos \theta }$	${\displaystyle \mp \sin \theta }$	${\displaystyle -\cos \theta }$	${\displaystyle \pm \sin \theta }$	${\displaystyle {\phantom {-}}\cos \theta }$
${\displaystyle \sec {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle {\phantom {-}}\sec \theta }$	${\displaystyle \mp \csc \theta }$	${\displaystyle -\sec \theta }$	${\displaystyle \pm \csc \theta }$	${\displaystyle {\phantom {-}}\sec \theta }$
${\displaystyle \tan {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle -\tan \theta }$	${\displaystyle \mp \cot \theta }$	${\displaystyle \pm \tan \theta }$	${\displaystyle \mp \cot \theta }$	${\displaystyle \pm \tan \theta }$
${\displaystyle \cot {\underline {\;~~~~~~~~~~~~~~\;}}=}$	${\displaystyle -\cot \theta }$	${\displaystyle \mp \tan \theta }$	${\displaystyle \pm \cot \theta }$	${\displaystyle \mp \tan \theta }$	${\displaystyle \pm \cot \theta }$

These formulas imply, in particular, that the following hold:

$${\displaystyle {\begin{aligned}\sin \theta &=-\sin(-\theta )&&=-\sin(\pi +\theta )&&={\phantom {-}}\sin(\pi -\theta )\\&=-\cos \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cos \left({\frac {\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {\pi }{2}}-\theta \right)\\&={\phantom {-}}\cos \left(-{\frac {\pi }{2}}+\theta \right)&&=-\cos \left({\frac {3\pi }{2}}-\theta \right)&&=-\cos \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\cos \theta &={\phantom {-}}\cos(-\theta )&&=-\cos(\pi +\theta )&&={\phantom {-}}\cos(\pi -\theta )\\&={\phantom {-}}\sin \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\sin \left({\frac {\pi }{2}}-\theta \right)&&=-\sin \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\sin \left(-{\frac {\pi }{2}}+\theta \right)&&=-\sin \left({\frac {3\pi }{2}}-\theta \right)&&={\phantom {-}}\sin \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\tan \theta &=-\tan(-\theta )&&={\phantom {-}}\tan(\pi +\theta )&&=-\tan(\pi -\theta )\\&=-\cot \left({\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {\pi }{2}}-\theta \right)&&={\phantom {-}}\cot \left(-{\frac {\pi }{2}}-\theta \right)\\&=-\cot \left(-{\frac {\pi }{2}}+\theta \right)&&={\phantom {-}}\cot \left({\frac {3\pi }{2}}-\theta \right)&&=-\cot \left(-{\frac {3\pi }{2}}+\theta \right)\\[0.3ex]\end{aligned}}}$$

where swapping ${\displaystyle \sin \leftrightarrow \csc ,}$ swapping ${\displaystyle \cos \leftrightarrow \sec ,}$ and swapping ${\displaystyle \tan \leftrightarrow \cot }$ gives the analogous equations for ${\displaystyle \csc ,\sec ,{\text{ and }}\cot ,}$ respectively.

So for example, by using the equality ${\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=\cos \theta ,}$ the equation ${\displaystyle \cos \theta =x}$ can be transformed into ${\textstyle \sin \left({\frac {\pi }{2}}-\theta \right)=x,}$ which allows for the solution to the equation ${\displaystyle \;\sin \varphi =x\;}$ (where ${\textstyle \varphi$ :={\frac {\pi }{2}}-\theta } ) to be used; that solution being: ${\displaystyle \varphi =(-1)^{k}\arcsin(x)+\pi k\;{\text{ for some }}k\in \mathbb {Z} ,}$ which becomes:

$${\displaystyle {\frac {\pi }{2}}-\theta ~=~(-1)^{k}\arcsin(x)+\pi k\quad {\text{ for some }}k\in \mathbb {Z} }$$

where using the fact that ${\displaystyle (-1)^{k}=(-1)^{-k}}$ and substituting ${\displaystyle h:=-k}$ proves that another solution to ${\displaystyle \;\cos \theta =x\;}$ is:

$${\displaystyle \theta ~=~(-1)^{h+1}\arcsin(x)+\pi h+{\frac {\pi }{2}}\quad {\text{ for some }}h\in \mathbb {Z} .}$$

The substitution ${\displaystyle \;\arcsin x={\frac {\pi }{2}}-\arccos x\;}$ may be used express the right hand side of the above formula in terms of ${\displaystyle \;\arccos x\;}$ instead of ${\displaystyle \;\arcsin x.\;}$

The table below shows how two angles ${\displaystyle \theta }$ and ${\displaystyle \varphi }$ must be related if their values under a given trigonometric function are equal or negatives of each other.

Equation	if and only if	Solution (for some ${\displaystyle k\in \mathbb {Z} }$)	Also a solution to
${\displaystyle {\phantom {-}}\sin \theta =\sin \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\csc \theta =\csc \varphi }$
${\displaystyle {\phantom {-}}\cos \theta =\cos \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\sec \theta =\sec \varphi }$
${\displaystyle {\phantom {-}}\tan \theta =\tan \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\cot \theta =\cot \varphi }$
${\displaystyle -\sin \theta =\sin \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle -\csc \theta =\csc \varphi }$
${\displaystyle -\cos \theta =\cos \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}}$	${\displaystyle -\sec \theta =\sec \varphi }$
${\displaystyle -\tan \theta =\tan \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle -\cot \theta =\cot \varphi }$
${\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}$

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is by considering the geometry of a right-angled triangle, with one side of length 1 and another side of length ${\displaystyle x,}$ then applying the Pythagorean theorem and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that ${\displaystyle x}$ is positive, and thus the result has to be corrected through the use of absolute values and the signum (sgn) operation.

${\displaystyle \theta }$	${\displaystyle \sin(\theta )}$	${\displaystyle \cos(\theta )}$	${\displaystyle \tan(\theta )}$	Diagram
${\displaystyle \arcsin(x)}$	${\displaystyle \sin(\arcsin(x))=x}$	${\displaystyle \cos(\arcsin(x))={\sqrt {1-x^{2}}}}$	${\displaystyle \tan(\arcsin(x))={\frac {x}{\sqrt {1-x^{2}}}}}$
${\displaystyle \arccos(x)}$	${\displaystyle \sin(\arccos(x))={\sqrt {1-x^{2}}}}$	${\displaystyle \cos(\arccos(x))=x}$	${\displaystyle \tan(\arccos(x))={\frac {\sqrt {1-x^{2}}}{x}}}$
${\displaystyle \arctan(x)}$	${\displaystyle \sin(\arctan(x))={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\arctan(x))={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \tan(\arctan(x))=x}$
${\displaystyle \operatorname {arccot}(x)}$	${\displaystyle \sin(\operatorname {arccot}(x))={\frac {1}{\sqrt {1+x^{2}}}}}$	${\displaystyle \cos(\operatorname {arccot}(x))={\frac {x}{\sqrt {1+x^{2}}}}}$	${\displaystyle \tan(\operatorname {arccot}(x))={\frac {1}{x}}}$
${\displaystyle \operatorname {arcsec}(x)}$	${\displaystyle \sin(\operatorname {arcsec}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}$	${\displaystyle \cos(\operatorname {arcsec}(x))={\frac {1}{x}}}$	${\displaystyle \tan(\operatorname {arcsec}(x))=\operatorname {sgn}(x){\sqrt {x^{2}-1}}}$
${\displaystyle \operatorname {arccsc}(x)}$	${\displaystyle \sin(\operatorname {arccsc}(x))={\frac {1}{x}}}$	${\displaystyle \cos(\operatorname {arccsc}(x))={\frac {\sqrt {x^{2}-1}}{|x|}}}$	${\displaystyle \tan(\operatorname {arccsc}(x))={\frac {\operatorname {sgn}(x)}{\sqrt {x^{2}-1}}}}$