Inverse trigonometric functions

"Arctangent" redirects here. For the music event, see ArcTanGent Festival.

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions,^{[1][2][3][4][5]} antitrigonometric functions^[6] or cyclometric functions^[7][8][9]) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions,^[10] and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

Notation

 

For a circle of radius 1, arcsin and arccos are the lengths of actual arcs determined by the quantities in question.

See also: Trigonometric functions § Notation

Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc.^[6] (This convention is used throughout this article.) This notation arises from the following geometric relationships:^{[citation needed]} when measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians.^[11] In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms asin, acos, atan.^[12]

The notations sin⁻¹(x), cos⁻¹(x), tan⁻¹(x), etc., as introduced by John Herschel in 1813,^[13][14] are often used as well in English-language sources,^[6] much more than the also established sin^[−1](x), cos^[−1](x), tan^[−1](x) – conventions consistent with the notation of an inverse function, that is useful (for example) to define the multivalued version of each inverse trigonometric function: ${\displaystyle \tan ^{-1}(x)=\{\arctan(x)+\pi k\mid k\in \mathbb {Z} \}~.}$  However, this might appear to conflict logically with the common semantics for expressions such as sin²(x) (although only sin² x, without parentheses, is the really common use), which refer to numeric power rather than function composition, and therefore may result in confusion between notation for the reciprocal (multiplicative inverse) and inverse function.^[15]

The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, (cos(x))⁻¹ = sec(x). Nevertheless, certain authors advise against using it, since it is ambiguous.^[6][16] Another precarious convention used by a small number of authors is to use an uppercase first letter, along with a “−1” superscript: Sin⁻¹(x), Cos⁻¹(x), Tan⁻¹(x), etc.^[17] Although it is intended to avoid confusion with the reciprocal, which should be represented by sin⁻¹(x), cos⁻¹(x), etc., or, better, by sin⁻¹ x, cos⁻¹ x, etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g. Mathematica, and MAGMA) use those very same capitalised representations for the standard trig functions, whereas others (Python, SymPy, NumPy, Matlab, MAPLE, etc.) use lower-case.

Hence, since 2009, the ISO 80000-2 standard has specified solely the "arc" prefix for the inverse functions.

Basic concepts

Principal values

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^{[citation needed]} 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 ${\displaystyle \cos \theta \neq 0,}$ 

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

 

Solutions to elementary trigonometric equations

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 false), because otherwise "LHS ${\displaystyle \,\iff \,}$  RHS" would not have been written (see this footnote^{[note 1]} for an example illustrating this concept).

Solutions to elementary trigonometric functions

Equation	if and only if	Solution ​(for some ${\displaystyle k\in \mathbb {Z} }$ )	if and only if	Expanded form of solution ​(for some ${\displaystyle k\in \mathbb {Z} }$ )
${\displaystyle \sin \theta =y}$	${\displaystyle \iff }$	${\displaystyle \theta =(-1)^{k}\arcsin(y)+{\phantom {2\,}}\pi k}$	${\displaystyle \iff }$	${\displaystyle {\begin{aligned}\theta &={\phantom {-}}\arcsin(y)+2\pi h\\&=-\arcsin(y)+2\pi h+\pi \end{aligned}}}$
${\displaystyle \csc \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta =(-1)^{k}\operatorname {arccsc} (r)+{\phantom {2\,}}\pi k}$	${\displaystyle \iff }$	${\displaystyle {\begin{aligned}\theta &={\phantom {-}}\operatorname {arccsc}(y)+2\pi h\\&=-\operatorname {arccsc}(y)+2\pi h+\pi \end{aligned}}}$
${\displaystyle \cos \theta =x}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {\quad \,\,}}\pm \arccos(x)+2\,\pi k}$	${\displaystyle \iff }$	${\displaystyle {\begin{aligned}\theta &={\phantom {-}}\arccos(y)+2\pi h\\&=-\arccos(y)+2\pi h{\phantom {+\pi }}\end{aligned}}}$
${\displaystyle \sec \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {\quad \,\,}}\pm \,\operatorname {arcsec} (r)+2\,\pi k}$	${\displaystyle \iff }$	${\displaystyle {\begin{aligned}\theta &={\phantom {-}}\operatorname {arcsec}(y)+2\pi h\\&=-\operatorname {arcsec}(y)+2\pi h{\phantom {+\pi }}\end{aligned}}}$
${\displaystyle \tan \theta =s}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {(-1)\,\,}}\arctan(s)+{\phantom {2\,}}\pi k}$
${\displaystyle \cot \theta =r}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {(-1)\,\,\,}}\operatorname {arccot} (r)+{\phantom {2\,}}\pi k}$

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 ${\displaystyle y}$ ).

Detailed example and explanation of the "plus or minus" symbol ±

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 ${\displaystyle \theta =-\pi /2}$ ).

Transforming equations

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 ${\displaystyle \;\arcsin x.\;}$ 

Equal identical trigonometric functions

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
${\displaystyle {\phantom {-}}\sin \theta =\sin \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {\quad }}(-1)^{k}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\csc \theta =\csc \varphi }$
${\displaystyle {\phantom {-}}\cos \theta =\cos \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\sec \theta =\sec \varphi }$
${\displaystyle {\phantom {-}}\tan \theta =\tan \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {(-1)^{k+1}}}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\phantom {-}}\cot \theta =\cot \varphi }$
${\displaystyle -\sin \theta =\sin \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta =(-1)^{k+1}\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle -\csc \theta =\csc \varphi }$
${\displaystyle -\cos \theta =\cos \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +2\pi k+\pi {\phantom {+\pi }}}$	${\displaystyle -\sec \theta =\sec \varphi }$
${\displaystyle -\tan \theta =\tan \varphi }$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}-\varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle -\cot \theta =\cot \varphi }$
${\displaystyle {\begin{aligned}{\phantom {-}}\left|\sin \theta \right|&=\left|\sin \varphi \right|\\{\phantom {-}}\left|\cos \theta \right|&=\left|\cos \varphi \right|\end{aligned}}}$	${\displaystyle \iff }$	${\displaystyle \theta ={\phantom {-1\quad }}\pm \varphi +{\phantom {2}}\pi k{\phantom {+\pi }}}$	${\displaystyle {\begin{aligned}{\phantom {-}}\left|\tan \theta \right|&=\left|\tan \varphi \right|\\\left|\csc \theta \right|&=\left|\csc \varphi \right|\\\left|\sec \theta \right|&=\left|\sec \varphi \right|\\\left|\cot \theta \right|&=\left|\cot \varphi \right|\end{aligned}}}$

Relationships between trigonometric functions and inverse trigonometric functions

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.

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Arctangent redirects here For the music event see ArcTanGent Festival In mathematics the inverse trigonometric functions occasionally also called arcus functions 1 2 3 4 5 antitrigonometric functions 6 or cyclometric functions 7 8 9 are the inverse functions of the trigonometric functions with suitably restricted domains Specifically they are the inverses of the sine cosine tangent cotangent secant and cosecant functions 10 and are used to obtain an angle from any of the angle s trigonometric ratios Inverse trigonometric functions are widely used in engineering navigation physics and geometry Contents 1 Notation 2 Basic concepts 2 1 Principal values 2 2 Solutions to elementary trigonometric equations 2 2 1 Detailed example and explanation of the plus or minus symbol 2 2 2 Transforming equations 2 2 3 Equal identical trigonometric functions 2 3 Relationships between trigonometric functions and inverse trigonometric functions 2 4 Relationships among the inverse trigonometric functions 2 5 Arctangent addition formula 3 In calculus 3 1 Derivatives of inverse trigonometric functions 3 2 Expression as definite integrals 3 3 Infinite series 3 3 1 Continued fractions for arctangent 3 4 Indefinite integrals of inverse trigonometric functions 3 4 1 Example 4 Extension to the complex plane 4 1 Logarithmic forms 4 1 1 Generalization 4 1 2 Example proof 5 Applications 5 1 Finding the angle of a right triangle 5 2 In computer science and engineering 5 2 1 Two argument variant of arctangent 5 2 2 Arctangent function with location parameter 5 2 3 Numerical accuracy 6 See also 7 Notes 8 References 9 External linksNotation Edit For a circle of radius 1 arcsin and arccos are the lengths of actual arcs determined by the quantities in question See also Trigonometric functions Notation Several notations for the inverse trigonometric functions exist The most common convention is to name inverse trigonometric functions using an arc prefix arcsin x arccos x arctan x etc 6 This convention is used throughout this article This notation arises from the following geometric relationships citation needed when measuring in radians an angle of 8 radians will correspond to an arc whose length is r8 where r is the radius of the circle Thus in the unit circle the arc whose cosine is x is the same as the angle whose cosine is x because the length of the arc of the circle in radii is the same as the measurement of the angle in radians 11 In computer programming languages the inverse trigonometric functions are often called by the abbreviated forms asin acos atan 12 The notations sin 1 x cos 1 x tan 1 x etc as introduced by John Herschel in 1813 13 14 are often used as well in English language sources 6 much more than the also established sin 1 x cos 1 x tan 1 x conventions consistent with the notation of an inverse function that is useful for example to define the multivalued version of each inverse trigonometric function tan 1 x arctan x p k k Z displaystyle tan 1 x arctan x pi k mid k in mathbb Z However this might appear to conflict logically with the common semantics for expressions such as sin2 x although only sin2 x without parentheses is the really common use which refer to numeric power rather than function composition and therefore may result in confusion between notation for the reciprocal multiplicative inverse and inverse function 15 The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name for example cos x 1 sec x Nevertheless certain authors advise against using it since it is ambiguous 6 16 Another precarious convention used by a small number of authors is to use an uppercase first letter along with a 1 superscript Sin 1 x Cos 1 x Tan 1 x etc 17 Although it is intended to avoid confusion with the reciprocal which should be represented by sin 1 x cos 1 x etc or better by sin 1 x cos 1 x etc it in turn creates yet another major source of ambiguity especially since many popular high level programming languages e g Mathematica and MAGMA use those very same capitalised representations for the standard trig functions whereas others Python SymPy NumPy Matlab MAPLE etc use lower case Hence since 2009 the ISO 80000 2 standard has specified solely the arc prefix for the inverse functions Basic concepts EditPrincipal values Edit 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 y x displaystyle y sqrt x could be defined from y 2 x displaystyle y 2 x the function y arcsin x displaystyle y arcsin x is defined so that sin y x displaystyle sin y x For a given real number x displaystyle x with 1 x 1 displaystyle 1 leq x leq 1 there are multiple in fact countably infinitely many numbers y displaystyle y such that sin y x displaystyle sin y x for example sin 0 0 displaystyle sin 0 0 but also sin p 0 displaystyle sin pi 0 sin 2 p 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 x displaystyle x in the domain the expression arcsin x 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 x displaystyle x for real result Range of usual principal value radians Range of usual principal value degrees arcsine y arcsin x displaystyle y arcsin x x sin y 1 x 1 displaystyle 1 leq x leq 1 p 2 y p 2 displaystyle frac pi 2 leq y leq frac pi 2 90 y 90 displaystyle 90 circ leq y leq 90 circ arccosine y arccos x displaystyle y arccos x x cos y 1 x 1 displaystyle 1 leq x leq 1 0 y p displaystyle 0 leq y leq pi 0 y 180 displaystyle 0 circ leq y leq 180 circ arctangent y arctan x displaystyle y arctan x x tan y all real numbers p 2 lt y lt p 2 displaystyle frac pi 2 lt y lt frac pi 2 90 lt y lt 90 displaystyle 90 circ lt y lt 90 circ arccotangent y arccot x displaystyle y operatorname arccot x x cot y all real numbers 0 lt y lt p displaystyle 0 lt y lt pi 0 lt y lt 180 displaystyle 0 circ lt y lt 180 circ arcsecant y arcsec x displaystyle y operatorname arcsec x x sec y x 1 displaystyle left vert x right vert geq 1 0 y lt p 2 or p 2 lt y p displaystyle 0 leq y lt frac pi 2 text or frac pi 2 lt y leq pi 0 y lt 90 or 90 lt y 180 displaystyle 0 circ leq y lt 90 circ text or 90 circ lt y leq 180 circ arccosecant y arccsc x displaystyle y operatorname arccsc x x csc y x 1 displaystyle left vert x right vert geq 1 p 2 y lt 0 or 0 lt y p 2 displaystyle frac pi 2 leq y lt 0 text or 0 lt y leq frac pi 2 90 y lt 0 or 0 lt y 90 displaystyle 90 circ leq y lt 0 circ text or 0 circ lt y leq 90 circ Note Some authors citation needed define the range of arcsecant to be 0 y lt p 2 or p y lt 3 p 2 textstyle 0 leq y lt frac pi 2 text or pi leq y lt frac 3 pi 2 because the tangent function is nonnegative on this domain This makes some computations more consistent For example using this range tan arcsec x x 2 1 displaystyle tan operatorname arcsec x sqrt x 2 1 whereas with the range 0 y lt p 2 or p 2 lt y p textstyle 0 leq y lt frac pi 2 text or frac pi 2 lt y leq pi we would have to write tan arcsec x x 2 1 displaystyle tan operatorname arcsec x pm sqrt x 2 1 since tangent is nonnegative on 0 y lt p 2 textstyle 0 leq y lt frac pi 2 but nonpositive on p 2 lt y p textstyle frac pi 2 lt y leq pi For a similar reason the same authors define the range of arccosecant to be p lt y p 2 textstyle pi lt y leq frac pi 2 or 0 lt y p 2 textstyle 0 lt y leq frac pi 2 If x displaystyle x is allowed to be a complex number then the range of y 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 valuessine sin displaystyle sin displaystyle R displaystyle mathbb R displaystyle to 1 1 displaystyle 1 1 arcsin displaystyle arcsin displaystyle 1 1 displaystyle 1 1 displaystyle to p 2 p 2 displaystyle left tfrac pi 2 tfrac pi 2 right cosine cos displaystyle cos displaystyle R displaystyle mathbb R displaystyle to 1 1 displaystyle 1 1 arccos displaystyle arccos displaystyle 1 1 displaystyle 1 1 displaystyle to 0 p displaystyle 0 pi tangent tan displaystyle tan displaystyle p Z p 2 p 2 displaystyle pi mathbb Z left tfrac pi 2 tfrac pi 2 right displaystyle to R displaystyle mathbb R arctan displaystyle arctan displaystyle R displaystyle mathbb R displaystyle to p 2 p 2 displaystyle left tfrac pi 2 tfrac pi 2 right cotangent cot displaystyle cot displaystyle p Z 0 p displaystyle pi mathbb Z 0 pi displaystyle to R displaystyle mathbb R arccot displaystyle operatorname arccot displaystyle R displaystyle mathbb R displaystyle to 0 p displaystyle 0 pi secant sec displaystyle sec displaystyle p Z p 2 p 2 displaystyle pi mathbb Z left tfrac pi 2 tfrac pi 2 right displaystyle to R 1 1 displaystyle mathbb R setminus 1 1 arcsec displaystyle operatorname arcsec displaystyle R 1 1 displaystyle mathbb R setminus 1 1 displaystyle to 0 p p 2 displaystyle 0 pi setminus left tfrac pi 2 right cosecant csc displaystyle csc displaystyle p Z 0 p displaystyle pi mathbb Z 0 pi displaystyle to R 1 1 displaystyle mathbb R setminus 1 1 arccsc displaystyle operatorname arccsc displaystyle R 1 1 displaystyle mathbb R setminus 1 1 displaystyle to p 2 p 2 0 displaystyle left tfrac pi 2 tfrac pi 2 right setminus 0 The symbol R displaystyle mathbb R infty infty denotes the set of all real numbers and Z 2 1 0 1 2 displaystyle mathbb Z ldots 2 1 0 1 2 ldots denotes the set of all integers The set of all integer multiples of p displaystyle pi is denoted byp Z p n n Z 2 p p 0 p 2 p 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 R 1 1 1 1 displaystyle mathbb R setminus 1 1 infty 1 cup 1 infty is the set of points in R displaystyle mathbb R that is real numbers that are not in the interval 1 1 displaystyle 1 1 The Minkowski sum notation p Z 0 p textstyle pi mathbb Z 0 pi and p Z p 2 p 2 displaystyle pi mathbb Z bigl tfrac pi 2 tfrac pi 2 bigr that is used above to concisely write the domains of cot csc tan and sec displaystyle cot csc tan text and sec is now explained Domain of cotangent cot displaystyle cot and cosecant csc displaystyle csc The domains of cot displaystyle cot and csc displaystyle csc are the same They are the set of all angles 8 displaystyle theta at which sin 8 0 displaystyle sin theta neq 0 i e all real numbers that are not of the form p n displaystyle pi n for some integer n displaystyle n p Z 0 p 2 p p p 0 0 p p 2 p R p Z displaystyle begin aligned pi mathbb Z 0 pi amp cdots cup 2 pi pi cup pi 0 cup 0 pi cup pi 2 pi cup cdots amp mathbb R setminus pi mathbb Z end aligned Domain of tangent tan displaystyle tan and secant sec displaystyle sec The domains of tan displaystyle tan and sec displaystyle sec are the same They are the set of all angles 8 displaystyle theta at which cos 8 0 displaystyle cos theta neq 0 p Z p 2 p 2 3 p 2 p 2 p 2 p 2 p 2 3 p 2 R p 2 p Z displaystyle begin aligned pi mathbb Z left tfrac pi 2 tfrac pi 2 right amp 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 amp mathbb R setminus left tfrac pi 2 pi mathbb Z right end aligned Solutions to elementary trigonometric equations Edit Each of the trigonometric functions is periodic in the real part of its argument running through all its values twice in each interval of 2 p displaystyle 2 pi Sine and cosecant begin their period at 2 p k p 2 textstyle 2 pi k frac pi 2 where k displaystyle k is an integer finish it at 2 p k p 2 textstyle 2 pi k frac pi 2 and then reverse themselves over 2 p k p 2 textstyle 2 pi k frac pi 2 to 2 p k 3 p 2 textstyle 2 pi k frac 3 pi 2 Cosine and secant begin their period at 2 p k displaystyle 2 pi k finish it at 2 p k p displaystyle 2 pi k pi and then reverse themselves over 2 p k p displaystyle 2 pi k pi to 2 p k 2 p displaystyle 2 pi k 2 pi Tangent begins its period at 2 p k p 2 textstyle 2 pi k frac pi 2 finishes it at 2 p k p 2 textstyle 2 pi k frac pi 2 and then repeats it forward over 2 p k p 2 textstyle 2 pi k frac pi 2 to 2 p k 3 p 2 textstyle 2 pi k frac 3 pi 2 Cotangent begins its period at 2 p k displaystyle 2 pi k finishes it at 2 p k p displaystyle 2 pi k pi and then repeats it forward over 2 p k p displaystyle 2 pi k pi to 2 p k 2 p displaystyle 2 pi k 2 pi This periodicity is reflected in the general inverses where k 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 8 displaystyle theta r displaystyle r s displaystyle s x displaystyle x and y displaystyle y all lie within appropriate ranges so that the relevant expressions below are well defined Note that for some k Z displaystyle k in mathbb Z is just another way of saying for some integer k 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 false because otherwise LHS displaystyle iff RHS would not have been written see this footnote note 1 for an example illustrating this concept Solutions to elementary trigonometric functions Equation if and only if Solution wbr for some k Z displaystyle k in mathbb Z if and only if Expanded form of solution wbr for some k Z displaystyle k in mathbb Z sin 8 y displaystyle sin theta y displaystyle iff 8 1 k arcsin y 2 p k displaystyle theta 1 k arcsin y phantom 2 pi k displaystyle iff 8 arcsin y 2 p h arcsin y 2 p h p displaystyle begin aligned theta amp phantom arcsin y 2 pi h amp arcsin y 2 pi h pi end aligned csc 8 r displaystyle csc theta r displaystyle iff 8 1 k arccsc r 2 p k displaystyle theta 1 k operatorname arccsc r phantom 2 pi k displaystyle iff 8 arccsc y 2 p h arccsc y 2 p h p displaystyle begin aligned theta amp phantom operatorname arccsc y 2 pi h amp operatorname arccsc y 2 pi h pi end aligned cos 8 x displaystyle cos theta x displaystyle iff 8 arccos x 2 p k displaystyle theta phantom quad pm arccos x 2 pi k displaystyle iff 8 arccos y 2 p h arccos y 2 p h p displaystyle begin aligned theta amp phantom arccos y 2 pi h amp arccos y 2 pi h phantom pi end aligned sec 8 r displaystyle sec theta r displaystyle iff 8 arcsec r 2 p k displaystyle theta phantom quad pm operatorname arcsec r 2 pi k displaystyle iff 8 arcsec y 2 p h arcsec y 2 p h p displaystyle begin aligned theta amp phantom operatorname arcsec y 2 pi h amp operatorname arcsec y 2 pi h phantom pi end aligned tan 8 s displaystyle tan theta s displaystyle iff 8 1 arctan s 2 p k displaystyle theta phantom 1 arctan s phantom 2 pi k cot 8 r displaystyle cot theta r displaystyle iff 8 1 arccot r 2 p k displaystyle theta phantom 1 operatorname arccot r phantom 2 pi k For example if cos 8 1 displaystyle cos theta 1 then 8 p 2 p k p 2 p 1 k displaystyle theta pi 2 pi k pi 2 pi 1 k for some k Z displaystyle k in mathbb Z While if sin 8 1 displaystyle sin theta pm 1 then 8 p 2 p k p 2 p k 1 textstyle theta frac pi 2 pi k frac pi 2 pi k 1 for some k Z displaystyle k in mathbb Z where k displaystyle k will be even if sin 8 1 displaystyle sin theta 1 and it will be odd if sin 8 1 displaystyle sin theta 1 The equations sec 8 1 displaystyle sec theta 1 and csc 8 1 displaystyle csc theta pm 1 have the same solutions as cos 8 1 displaystyle cos theta 1 and sin 8 1 displaystyle sin theta pm 1 respectively In all equations above except for those just solved i e except for sin displaystyle sin csc 8 1 displaystyle csc theta pm 1 and cos displaystyle cos sec 8 1 displaystyle sec theta 1 the integer k displaystyle k in the solution s formula is uniquely determined by 8 displaystyle theta for fixed r s x displaystyle r s x and y displaystyle y Detailed example and explanation of the plus or minus symbol Edit The solutions to cos 8 x displaystyle cos theta x and sec 8 x displaystyle sec theta x involve the plus or minus symbol displaystyle pm whose meaning is now clarified Only the solution to cos 8 x displaystyle cos theta x will be discussed since the discussion for sec 8 x displaystyle sec theta x is the same We are given x displaystyle x between 1 x 1 displaystyle 1 leq x leq 1 and we know that there is an angle 8 displaystyle theta in some interval that satisfies cos 8 x displaystyle cos theta x We want to find this 8 displaystyle theta The table above indicates that the solution is8 arccos x 2 p k for some k Z 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 8 arccos x 2 p k displaystyle theta arccos x 2 pi k for some integer k displaystyle k or 8 arccos x 2 p k displaystyle theta arccos x 2 pi k for some integer k displaystyle k As mentioned above if arccos x p displaystyle arccos x pi which by definition only happens when x cos p 1 displaystyle x cos pi 1 then both statements 1 and 2 hold although with different values for the integer k displaystyle k if K displaystyle K is the integer from statement 1 meaning that 8 p 2 p K displaystyle theta pi 2 pi K holds then the integer k displaystyle k for statement 2 is K 1 displaystyle K 1 because 8 p 2 p 1 K displaystyle theta pi 2 pi 1 K However if x 1 displaystyle x neq 1 then the integer k displaystyle k is unique and completely determined by 8 displaystyle theta If arccos x 0 displaystyle arccos x 0 which by definition only happens when x cos 0 1 displaystyle x cos 0 1 then arccos x 0 displaystyle pm arccos x 0 because arccos x 0 0 displaystyle arccos x 0 0 and arccos x 0 0 displaystyle arccos x 0 0 so in both cases arccos x displaystyle pm arccos x is equal to 0 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 arccos x 0 displaystyle arccos x 0 and arccos x p displaystyle arccos x pi we now focus on the case where arccos x 0 displaystyle arccos x neq 0 and arccos x p displaystyle arccos x neq pi So assume this from now on The solution to cos 8 x displaystyle cos theta x is still8 arccos x 2 p k for some k Z 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 arccos x 0 displaystyle arccos x neq 0 and 0 lt arccos x lt p displaystyle 0 lt arccos x lt pi statements 1 and 2 are different and furthermore exactly one of the two equalities holds not both Additional information about 8 displaystyle theta is needed to determine which one holds For example suppose that x 0 displaystyle x 0 and that all that is known about 8 displaystyle theta is that p 8 p displaystyle pi leq theta leq pi and nothing more is known Then arccos x arccos 0 p 2 displaystyle arccos x arccos 0 frac pi 2 and moreover in this particular case k 0 displaystyle k 0 for both the displaystyle case and the displaystyle case and so consequently 8 arccos x 2 p k p 2 2 p 0 p 2 displaystyle theta pm arccos x 2 pi k pm left frac pi 2 right 2 pi 0 pm frac pi 2 This means that 8 displaystyle theta could be either p 2 displaystyle pi 2 or p 2 displaystyle pi 2 Without additional information it is not possible to determine which of these values 8 displaystyle theta has An example of some additional information that could determine the value of 8 displaystyle theta would be knowing that the angle is above the x displaystyle x axis in which case 8 p 2 displaystyle theta pi 2 or alternatively knowing that it is below the x displaystyle x axis in which case 8 p 2 displaystyle theta pi 2 Transforming equations Edit The equations above can be transformed by using the reflection and shift identities 18 Transforming equations by shifts and reflections Argument displaystyle underline 8 displaystyle theta p 2 8 displaystyle frac pi 2 pm theta p 8 displaystyle pi pm theta 3 p 2 8 displaystyle frac 3 pi 2 pm theta 2 k p 8 displaystyle 2k pi pm theta k Z displaystyle k in mathbb Z sin displaystyle sin underline sin 8 displaystyle sin theta cos 8 displaystyle phantom cos theta sin 8 displaystyle mp sin theta cos 8 displaystyle cos theta sin 8 displaystyle pm sin theta csc displaystyle csc underline csc 8 displaystyle csc theta sec 8 displaystyle phantom sec theta csc 8 displaystyle mp csc theta sec 8 displaystyle sec theta csc 8 displaystyle pm csc theta cos displaystyle cos underline cos 8 displaystyle phantom cos theta sin 8 displaystyle mp sin theta cos 8 displaystyle cos theta sin 8 displaystyle pm sin theta cos 8 displaystyle phantom cos theta sec displaystyle sec underline sec 8 displaystyle phantom sec theta csc 8 displaystyle mp csc theta sec 8 displaystyle sec theta csc 8 displaystyle pm csc theta sec 8 displaystyle phantom sec theta tan displaystyle tan underline tan 8 displaystyle tan theta cot 8 displaystyle mp cot theta tan 8 displaystyle pm tan theta cot 8 displaystyle mp cot theta tan 8 displaystyle pm tan theta cot displaystyle cot underline cot 8 displaystyle cot theta tan 8 displaystyle mp tan theta cot 8 displaystyle pm cot theta tan 8 displaystyle mp tan theta cot 8 displaystyle pm cot theta These formulas imply in particular that the following hold sin 8 sin 8 sin p 8 sin p 8 cos p 2 8 cos p 2 8 cos p 2 8 cos p 2 8 cos 3 p 2 8 cos 3 p 2 8 cos 8 cos 8 cos p 8 cos p 8 sin p 2 8 sin p 2 8 sin p 2 8 sin p 2 8 sin 3 p 2 8 sin 3 p 2 8 tan 8 tan 8 tan p 8 tan p 8 cot p 2 8 cot p 2 8 cot p 2 8 cot p 2 8 cot 3 p 2 8 cot 3 p 2 8 displaystyle begin aligned sin theta amp sin theta amp amp sin pi theta amp amp phantom sin pi theta amp cos left frac pi 2 theta right amp amp phantom cos left frac pi 2 theta right amp amp cos left frac pi 2 theta right amp phantom cos left frac pi 2 theta right amp amp cos left frac 3 pi 2 theta right amp amp cos left frac 3 pi 2 theta right 0 3ex cos theta amp phantom cos theta amp amp cos pi theta amp amp phantom cos pi theta amp phantom sin left frac pi 2 theta right amp amp phantom sin left frac pi 2 theta right amp amp sin left frac pi 2 theta right amp sin left frac pi 2 theta right amp amp sin left frac 3 pi 2 theta right amp amp phantom sin left frac 3 pi 2 theta right 0 3ex tan theta amp tan theta amp amp phantom tan pi theta amp amp tan pi theta amp cot left frac pi 2 theta right amp amp phantom cot left frac pi 2 theta right amp amp phantom cot left frac pi 2 theta right amp cot left frac pi 2 theta right amp amp phantom cot left frac 3 pi 2 theta right amp amp cot left frac 3 pi 2 theta right 0 3ex end aligned where swapping sin csc displaystyle sin leftrightarrow csc swapping cos sec displaystyle cos leftrightarrow sec and swapping tan cot displaystyle tan leftrightarrow cot gives the analogous equations for csc sec and cot displaystyle csc sec text and cot respectively So for example by using the equality sin p 2 8 cos 8 textstyle sin left frac pi 2 theta right cos theta the equation cos 8 x displaystyle cos theta x can be transformed into sin p 2 8 x textstyle sin left frac pi 2 theta right x which allows for the solution to the equation sin f x displaystyle sin varphi x where f p 2 8 textstyle varphi frac pi 2 theta to be used that solution being f 1 k arcsin x p k for some k Z displaystyle varphi 1 k arcsin x pi k text for some k in mathbb Z which becomes p 2 8 1 k arcsin x p k for some k Z displaystyle frac pi 2 theta 1 k arcsin x pi k quad text for some k in mathbb Z where using the fact that 1 k 1 k displaystyle 1 k 1 k and substituting h k displaystyle h k proves that another solution to cos 8 x displaystyle cos theta x is 8 1 h 1 arcsin x p h p 2 for some h Z displaystyle theta 1 h 1 arcsin x pi h frac pi 2 quad text for some h in mathbb Z The substitution arcsin x p 2 arccos x displaystyle arcsin x frac pi 2 arccos x may be used express the right hand side of the above formula in terms of arccos x displaystyle arccos x instead of arcsin x displaystyle arcsin x Equal identical trigonometric functions Edit The table below shows how two angles 8 displaystyle theta and f 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 k Z displaystyle k in mathbb Z Also a solution to sin 8 sin f displaystyle phantom sin theta sin varphi displaystyle iff 8 1 k f 2 p k p displaystyle theta phantom quad 1 k varphi phantom 2 pi k phantom pi csc 8 csc f displaystyle phantom csc theta csc varphi cos 8 cos f displaystyle phantom cos theta cos varphi displaystyle iff 8 1 f 2 p k p displaystyle theta phantom 1 quad pm varphi 2 pi k phantom pi sec 8 sec f displaystyle phantom sec theta sec varphi tan 8 tan f displaystyle phantom tan theta tan varphi displaystyle iff 8 1 k 1 f 2 p k p displaystyle theta phantom 1 k 1 varphi phantom 2 pi k phantom pi cot 8 cot f displaystyle phantom cot theta cot varphi sin 8 sin f displaystyle sin theta sin varphi displaystyle iff 8 1 k 1 f 2 p k p displaystyle theta 1 k 1 varphi phantom 2 pi k phantom pi csc 8 csc f displaystyle csc theta csc varphi cos 8 cos f displaystyle cos theta cos varphi displaystyle iff 8 1 f 2 p k p p displaystyle theta phantom 1 quad pm varphi 2 pi k pi phantom pi sec 8 sec f displaystyle sec theta sec varphi tan 8 tan f displaystyle tan theta tan varphi displaystyle iff 8 1 f 2 p k p displaystyle theta phantom 1 quad varphi phantom 2 pi k phantom pi cot 8 cot f displaystyle cot theta cot varphi sin 8 sin f cos 8 cos f displaystyle begin aligned phantom left sin theta right amp left sin varphi right phantom left cos theta right amp left cos varphi right end aligned displaystyle iff 8 1 f 2 p k p displaystyle theta phantom 1 quad pm varphi phantom 2 pi k phantom pi tan 8 tan f csc 8 csc f sec 8 sec f cot 8 cot f displaystyle begin aligned phantom left tan theta right amp left tan varphi right left csc theta right amp left csc varphi right left sec theta right amp left sec varphi right left cot theta right amp left cot varphi right end aligned Relationships between trigonometric functions and inverse trigonometric functions Edit 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 x 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 x displaystyle x is positive and thus the result has to be corrected through the use of absolute values and the signum sgn operation 8 displaystyle theta sin 8 displaystyle sin theta cos 8 displaystyle cos theta span cla, wikipedia, wiki, book, books, library, article , read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games. © Copyright 2021, All Rights Reserved  |   www.wiki3.en-us.nina.az Rss Feed SiteMap Ping DMCA Contact Us HomePage Closed Call: