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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
s of the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
, under suitably restricted domains. Specifically, they are the inverses of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...
,
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
,
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
,
cotangent In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
,
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navig ...
,
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, and
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
.


Notation

Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of radians will correspond to an arc whose length is , where is the radius of the circle. Thus in the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "the arc whose cosine is " is the same as "the angle whose cosine is ", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages, the inverse trigonometric functions are often called by the abbreviated forms , , . The notations , , , etc., as introduced by
John Herschel Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor and experimental photographer who invented the blueprint and did botanical work. ...
in 1813, are often used as well in English-language sources, much more than the also established , , – conventions consistent with the notation of an
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
, that is useful (for example) to define the multivalued version of each inverse trigonometric function: \tan^(x) = \ ~. However, this might appear to conflict logically with the common semantics for expressions such as (although only , 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 In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
) and
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
. The confusion is somewhat mitigated by the fact that each of the reciprocal trigonometric functions has its own name — for example, . Nevertheless, certain authors advise against using it, since it is ambiguous. Another precarious convention used by a small number of authors is to use an
uppercase Letter case is the distinction between the letters that are in larger uppercase or capitals (more formally ''#Majuscule, majuscule'') and smaller lowercase (more formally ''#Minuscule, minuscule'') in the written representation of certain langua ...
first letter, along with a “” superscript: , , , etc. Although it is intended to avoid confusion with the reciprocal, which should be represented by , , etc., or, better, by , , etc., it in turn creates yet another major source of ambiguity, especially since many popular high-level programming languages (e.g.
Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
and
MAGMA Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...
) use those very same capitalised representations for the standard trig functions, whereas others ( Python, SymPy, NumPy,
Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
,
MAPLE ''Acer'' is a genus of trees and shrubs commonly known as maples. The genus is placed in the soapberry family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated si ...
, 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)
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
s of the domains of the original functions. For example, using in the sense of
multivalued function In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional p ...
s, just as the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
function y = \sqrt could be defined from y^2 = x, the function y = \arcsin(x) is defined so that \sin(y) = x. For a given real number x, with -1 \leq x \leq 1, there are multiple (in fact,
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
ly many) numbers y such that \sin(y) = x; for example, \sin(0) = 0, but also \sin(\pi) = 0, \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 in the domain, the expression \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. Note: Some authors define the range of arcsecant to be or because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, \tan(\arcsec(x)) = \sqrt, whereas with the range or we would have to write \tan(\arcsec(x)) = \pm \sqrt, since tangent is nonnegative on 0 \leq y < \frac, but nonpositive on \frac < y \leq \pi. For a similar reason, the same authors define the range of arccosecant to be ( - \pi < y \leq - \frac or 0 < y \leq \frac ) .


Domains

If is allowed to be a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
, then the range of applies only to its real part.


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 2 \pi: * Sine and cosecant begin their period at 2 \pi k-\frac (where k is an integer), finish it at 2 \pi k+\frac, and then reverse themselves over 2 \pi k+\frac to 2 \pi k+\frac. * Cosine and secant begin their period at 2 \pi k, finish it at 2 \pi k+\pi. and then reverse themselves over 2 \pi k+\pi to 2 \pi k+2 \pi. * Tangent begins its period at 2 \pi k-\frac, finishes it at 2 \pi k+\frac, and then repeats it (forward) over 2 \pi k+\frac to 2 \pi k+\frac. * Cotangent begins its period at 2 \pi k, finishes it at 2 \pi k+\pi, and then repeats it (forward) over 2 \pi k+\pi to 2 \pi k+2 \pi. This periodicity is reflected in the general inverses, where 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 \theta, r, s, x, and y all lie within appropriate ranges so that the relevant expressions below are
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression (mathematics), expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined ...
. Note that "for some k \in \Z" is just another way of saying "for some
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
k." The symbol \,\iff\, is logical equality and indicates that if the left hand side is true then so is the right hand side and, conversely, if the right hand side is true then so is the left hand side (see this footnoteThe expression "LHS \,\iff\, RHS" indicates that (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are true, or else (b) the left hand side and right hand side are false; there is option (c) (e.g. it is possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS \,\iff\, RHS" would not have been written.
To clarify, suppose that it is written "LHS \,\iff\, RHS" where LHS (which abbreviates ''left hand side'') and RHS are both statements that can individually be either be true or false. For example, if \theta and s are some given and fixed numbers and if the following is written: \tan \theta = s \,\iff\, \theta = \arctan(s)+\pi k \quad \text k \in \Z then LHS is the statement "\tan \theta = s". Depending on what specific values \theta and s have, this LHS statement can either be true or false. For instance, LHS is true if \theta = 0 and s = 0 (because in this case \tan \theta = \tan 0 = s) but LHS is false if \theta = 0 and s = 2 (because in this case \tan \theta = \tan 0 = s which is not equal to s = 2); more generally, LHS is false if \theta = 0 and s \neq 0. Similarly, RHS is the statement "\theta = \arctan(s)+\pi k for some k \in \Z". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values \theta and s have). The logical equality symbol \,\iff\, means that (a) if the LHS statement is true then the RHS statement is also true, and moreover (b) if the LHS statement is false then the RHS statement is also false. Similarly, \,\iff\, means that (c) if the RHS statement is true then the LHS statement is also true, and moreover (d) if the RHS statement is false then the LHS statement is also false.
for more details and an example illustrating this concept). where the first four solutions can be written in expanded form as: For example, if \cos \theta = -1 then \theta = \pi+2 \pi k = -\pi+2 \pi (1+k) for some k \in \Z. While if \sin \theta = \pm 1 then \theta = \frac+\pi k =-\frac+\pi (k+1) for some k \in \Z, where k will be even if \sin \theta = 1 and it will be odd if \sin \theta = -1. The equations \sec \theta = -1 and \csc \theta = \pm 1 have the same solutions as \cos \theta = -1 and \sin \theta = \pm 1, respectively. In all equations above for those just solved (i.e. except for \sin/\csc \theta = \pm 1 and \cos/\sec \theta =-1), the integer k in the solution's formula is uniquely determined by \theta (for fixed r, s, x, and y). With the help of integer parity \operatorname(h) = \begin 0 & \text h \text \\ 1 & \text h \text \\ \end it is possible to write a solution to \cos \theta = x that doesn't involve the "plus or minus" \,\pm\, symbol: :cos \; \theta = x \quad if and only if \quad \theta = (-1)^h \arccos(x) + \pi h + \pi \operatorname(h) \quad for some h \in \Z. And similarly for the secant function, :sec \; \theta = r \quad if and only if \quad \theta = (-1)^h \arcsec(r) + \pi h + \pi \operatorname(h) \quad for some h \in \Z, where \pi h + \pi \operatorname(h) equals \pi h when the integer h is even, and equals \pi h + \pi when it's odd.


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

The solutions to \cos \theta = x and \sec \theta = x involve the "plus or minus" symbol \,\pm,\, whose meaning is now clarified. Only the solution to \cos \theta = x will be discussed since the discussion for \sec \theta = x is the same. We are given x between -1 \leq x \leq 1 and we know that there is an angle \theta in some interval that satisfies \cos \theta = x. We want to find this \theta. The table above indicates that the solution is \,\theta = \pm \arccos x+2 \pi k\, \quad \textk \in \Z which is a shorthand way of saying that (at least) one of the following statement is true:
#\,\theta = \arccos x+2 \pi k\, for some integer k,
or #\,\theta =-\arccos x+2 \pi k\, for some integer k. As mentioned above, if \,\arccos x = \pi\, (which by definition only happens when x = \cos \pi = -1) then both statements (1) and (2) hold, although with different values for the integer k: if K is the integer from statement (1), meaning that \theta = \pi+2 \pi K holds, then the integer k for statement (2) is K+1 (because \theta = -\pi+2 \pi (1+K)). However, if x \neq -1 then the integer k is unique and completely determined by \theta. If \,\arccos x = 0\, (which by definition only happens when x = \cos 0 = 1) then \,\pm\arccos x = 0\, (because \,+ \arccos x = +0 = 0\, and \,-\arccos x = -0 = 0\, so in both cases \,\pm \arccos x\, is equal to 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\, and \,\arccos x = \pi,\, we now focus on the case where \,\arccos x \neq 0\, and \,\arccos x \neq \pi,\, So assume this from now on. The solution to \cos \theta = x is still \,\theta = \pm \arccos x+2 \pi k\, \quad \textk \in \Z which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because \,\arccos x \neq 0\, and \,0 < \arccos x < \pi,\, statements (1) and (2) are different and furthermore, ''exactly one'' of the two equalities holds (not both). Additional information about \theta is needed to determine which one holds. For example, suppose that x = 0 and that that is known about \theta is that \,-\pi \leq \theta \leq \pi\, (and nothing more is known). Then \arccos x = \arccos 0 = \frac and moreover, in this particular case k = 0 (for both the \,+\, case and the \,-\, case) and so consequently, \theta ~=~ \pm \arccos x+2 \pi k ~=~ \pm \left(\frac\right)+2\pi (0) ~=~ \pm \frac. This means that \theta could be either \,\pi/2\, or \,-\pi/2. Without additional information it is not possible to determine which of these values \theta has. An example of some additional information that could determine the value of \theta would be knowing that the angle is above the x-axis (in which case \theta = \pi/2) or alternatively, knowing that it is below the x-axis (in which case \theta =-\pi/2).


Equal identical trigonometric functions

;Set of all solutions to elementary trigonometric equations Thus given a single solution \theta to an elementary trigonometric equation (\sin \theta = y is such an equation, for instance, and because \sin (\arcsin y) = y always holds, \theta := \arcsin y is always a solution), the set of all solutions to it are:


Transforming equations

The equations above can be transformed by using the reflection and shift identities: These formulas imply, in particular, that the following hold: \begin \sin \theta &= -\sin(-\theta) &&= -\sin(\pi+\theta) &&= \phantom\sin(\pi-\theta) \\ &= -\cos\left(\frac+\theta\right) &&= \phantom\cos\left(\frac-\theta\right) &&= -\cos\left(-\frac-\theta\right) \\ &= \phantom\cos\left(-\frac+\theta\right) &&= -\cos\left(\frac-\theta\right) &&= -\cos\left(-\frac+\theta\right) \\ .3ex\cos \theta &= \phantom\cos(-\theta) &&= -\cos(\pi+\theta) &&= -\cos(\pi-\theta) \\ &= \phantom\sin\left(\frac+\theta\right) &&= \phantom\sin\left(\frac-\theta\right) &&= -\sin\left(-\frac-\theta\right) \\ &= -\sin\left(-\frac+\theta\right) &&= -\sin\left(\frac-\theta\right) &&= \phantom\sin\left(-\frac+\theta\right) \\ .3ex\tan \theta &= -\tan(-\theta) &&= \phantom\tan(\pi+\theta) &&= -\tan(\pi-\theta) \\ &= -\cot\left(\frac+\theta\right) &&= \phantom\cot\left(\frac-\theta\right) &&= \phantom\cot\left(-\frac-\theta\right) \\ &= -\cot\left(-\frac+\theta\right) &&= \phantom\cot\left(\frac-\theta\right) &&= -\cot\left(-\frac+\theta\right) \\ .3ex\end where swapping \sin \leftrightarrow \csc, swapping \cos \leftrightarrow \sec, and swapping \tan \leftrightarrow \cot gives the analogous equations for \csc, \sec, \text \cot, respectively. So for example, by using the equality \sin \left(\frac-\theta\right) = \cos \theta, the equation \cos \theta = x can be transformed into \sin \left(\frac-\theta\right) = x, which allows for the solution to the equation \;\sin \varphi = x\; (where \varphi := \frac-\theta) to be used; that solution being: \varphi = (-1)^k \arcsin (x)+\pi k \; \text k \in \Z, which becomes: \frac-\theta ~=~ (-1)^k \arcsin (x)+\pi k \quad \text k \in \Z where using the fact that (-1)^ = (-1)^ and substituting h :=-k proves that another solution to \;\cos \theta = x\; is: \theta ~=~ (-1)^ \arcsin (x)+\pi h+\frac \quad \text h \in \Z. The substitution \;\arcsin x = \frac-\arccos x\; may be used express the right hand side of the above formula in terms of \;\arccos x\; instead of \;\arcsin x.\;


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 x, then applying the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
and definitions of the trigonometric ratios. It is worth noting that for arcsecant and arccosecant, the diagram assumes that x is positive, and thus the result has to be corrected through the use of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
s and the signum (sgn) operation.


Relationships among the inverse trigonometric functions

Complementary angles: :\begin \arccos(x) &= \frac - \arcsin(x) \\ .5em\arccot(x) &= \frac - \arctan(x) \\ .5em\arccsc(x) &= \frac - \arcsec(x) \end Negative arguments: :\begin \arcsin(-x) &= -\arcsin(x) \\ \arccsc(-x) &= -\arccsc(x) \\ \arccos(-x) &= \pi -\arccos(x) \\ \arcsec(-x) &= \pi -\arcsec(x) \\ \arctan(-x) &= -\arctan(x) \\ \arccot(-x) &= \pi -\arccot(x) \end Reciprocal arguments: :\begin \arcsin\left(\frac\right) &= \arccsc(x) & \\ .3em\arccsc\left(\frac\right) &= \arcsin(x) & \\ .3em\arccos\left(\frac\right) &= \arcsec(x) & \\ .3em\arcsec\left(\frac\right) &= \arccos(x) & \\ .3em\arctan\left(\frac\right) &= \arccot(x) &= \frac - \arctan(x) \, , \text x > 0 \\ .3em\arctan\left(\frac\right) &= \arccot(x) - \pi &= -\frac - \arctan(x) \, , \text x < 0 \\ .3em\arccot\left(\frac\right) &= \arctan(x) &= \frac - \arccot(x) \, , \text x > 0 \\ .3em\arccot\left(\frac\right) &= \arctan(x) + \pi &= \frac - \arccot(x) \, , \text x < 0 \end The identities above can be used with (and derived from) the fact that \sin and \csc are reciprocals (i.e. \csc = \tfrac1), as are \cos and \sec, and \tan and \cot. Useful identities if one only has a fragment of a sine table: :\begin \arcsin(x) &= \frac\arccos\left(1-2x^2\right) \, , \text 0 \leq x \leq 1 \\ \arcsin(x) &= \arctan\left(\frac\right) \\ \arccos(x) &= \frac\arccos\left(2x^2-1\right) \, , \text 0 \leq x \leq 1 \\ \arccos(x) &= \arctan\left(\frac\right) \\ \arccos(x) &= \arcsin\left(\sqrt\right) \, , \text 0 \leq x \leq 1 \text \\ \arccos &\left(\frac\right) = \arcsin \left (\frac\right) \, , \text 0 \leq x \leq 1 \\ \arcsin &\left(\sqrt\right) =\frac-\sgn(x)\arcsin(x) \\ \arctan(x) &= \arcsin\left(\frac\right) \\ \arccot(x) &= \arccos\left(\frac\right) \end Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real). A useful form that follows directly from the table above is :\arctan(x) = \arccos\left(\sqrt\right)\, , \text x\geq 0 . It is obtained by recognizing that \cos\left(\arctan\left(x\right)\right) = \sqrt = \cos\left(\arccos\left(\sqrt\right)\right). From the half-angle formula, \tan\left(\tfrac\right) = \tfrac, we get: :\begin \arcsin(x) &= 2 \arctan\left(\frac\right) \\ .5em\arccos(x) &= 2 \arctan\left(\frac\right) \, , \text -1 < x \leq 1 \\ .5em\arctan(x) &= 2 \arctan\left(\frac\right) \end


Arctangent addition formula

:\arctan(u) \pm \arctan(v) = \arctan\left(\frac\right) \pmod \pi \, , \quad u v \ne 1 \, . This is derived from the tangent addition formula :\tan(\alpha \pm \beta) = \frac \, , by letting :\alpha = \arctan(u) \, , \quad \beta = \arctan(v) \, .


In calculus


Derivatives of inverse trigonometric functions

The
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s for complex values of ''z'' are as follows: :\begin \frac \arcsin(z) & = \frac \; ; &z &\neq -1, +1 \\ \frac \arccos(z) & = -\frac \; ; &z &\neq -1, +1 \\ \frac \arctan(z) & = \frac \; ; &z &\neq -i, +i\\ \frac \arccot(z) & = -\frac \; ; &z &\neq -i, +i \\ \frac \arcsec(z) & = \frac \; ; &z &\neq -1, 0, +1 \\ \frac \arccsc(z) & = -\frac \; ; &z &\neq -1, 0, +1 \end Only for real values of ''x'': :\begin \frac \arcsec(x) & = \frac \; ; & , x, > 1\\ \frac \arccsc(x) & = -\frac \; ; & , x, > 1 \end These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x = \sin \theta, then dx/d\theta = \cos \theta = \sqrt, so :\frac\arcsin(x) = \frac = \frac = \frac.


Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral: :\begin \arcsin(x) &= \int_0^x \frac \, dz \; , & , x, & \leq 1\\ \arccos(x) &= \int_x^1 \frac \, dz \; , & , x, & \leq 1\\ \arctan(x) &= \int_0^x \frac \, dz \; ,\\ \arccot(x) &= \int_x^\infty \frac \, dz \; ,\\ \arcsec(x) &= \int_1^x \frac \, dz = \pi + \int_^ \frac \, dz\; , & x & \geq 1\\ \arccsc(x) &= \int_x^\infty \frac \, dz = \int_^ \frac \, dz \; , & x & \geq 1\\ \end When ''x'' equals 1, the integrals with limited domains are
improper integral In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integral ...
s, but still well-defined.


Infinite series

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
, as follows. For arcsine, the series can be derived by expanding its derivative, \tfrac, as a
binomial series In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where \alpha is any complex number, and the power series on the right-hand side is expressed in terms of the ...
, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative \frac in a
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
, and applying the integral definition above (see Leibniz series). : \begin \arcsin(z) & = z + \left( \frac \right) \frac + \left( \frac \right) \frac + \left( \frac \right) \frac + \cdots \\ pt& = \sum_^\infty \frac\frac \\ pt& = \sum_^\infty \frac \frac \, ; \qquad , z, \le 1 \end :\arctan(z) = z - \frac +\frac - \frac + \cdots = \sum_^\infty \frac \, ; \qquad , z, \le 1 \qquad z \neq i,-i Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, \arccos(x) = \pi/2 - \arcsin(x), \arccsc(x) = \arcsin(1/x), and so on. Another series is given by: :2\left(\arcsin\left(\frac\right) \right)^2 = \sum_^\infty \frac.
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
found a series for the arctangent that converges more quickly than its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
: : \arctan(z) = \frac z \sum_^\infty \prod_^n \frac. (The term in the sum for ''n'' = 0 is the
empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplication, multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operat ...
, so is 1.) Alternatively, this can be expressed as :\arctan(z) = \sum_^\infty \frac \frac. Another series for the arctangent function is given by :\arctan(z) = i\sum_^\infty\frac\left(\frac - \frac\right), where i=\sqrt is the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
.


Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions: : \arctan(z) = \frac z = \frac The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (''nz'')2, with each perfect square appearing once. The first was developed by
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
; the second by
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
utilizing the Gaussian hypergeometric series.


Indefinite integrals of inverse trigonometric functions

For real and complex values of ''z'': :\begin \int \arcsin(z) \, dz &= z \, \arcsin(z) + \sqrt + C\\ \int \arccos(z) \, dz &= z \, \arccos(z) - \sqrt + C\\ \int \arctan(z) \, dz &= z \, \arctan(z) - \frac \ln \left( 1 + z^2 \right) + C\\ \int \arccot(z) \, dz &= z \, \arccot(z) + \frac \ln \left( 1 + z^2 \right) + C\\ \int \arcsec(z) \, dz &= z \, \arcsec(z) - \ln \left z \left( 1 + \sqrt \right) \right+ C\\ \int \arccsc(z) \, dz &= z \, \arccsc(z) + \ln \left z \left( 1 + \sqrt \right) \right+ C \end For real ''x'' ≥ 1: :\begin \int \arcsec(x) \, dx &= x \, \arcsec(x) - \ln \left( x + \sqrt \right) + C\\ \int \arccsc(x) \, dx &= x \, \arccsc(x) + \ln \left( x + \sqrt \right) + C \end For all real ''x'' not between -1 and 1: :\begin \int \arcsec(x) \, dx &= x \, \arcsec(x) - \sgn(x) \ln\left, x + \sqrt\ + C\\ \int \arccsc(x) \, dx &= x \, \arccsc(x) + \sgn(x) \ln\left, x + \sqrt\ + C \end The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecant functions. The signum function is also necessary due to the absolute values in the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s of the two functions, which create two different solutions for positive and negative values of x. These can be further simplified using the logarithmic definitions of the
inverse hyperbolic function In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangen ...
s: :\begin \int \arcsec(x) \, dx &= x \, \arcsec(x) - \operatorname(, x, ) + C\\ \int \arccsc(x) \, dx &= x \, \arccsc(x) + \operatorname(, x, ) + C\\ \end The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical to the signum logarithmic function shown above. All of these antiderivatives can be derived using
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
and the simple derivative forms shown above.


Example

Using \int u \, dv = u v - \int v \, du (i.e.
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivati ...
), set :\begin u &= \arcsin(x) & dv &= dx \\ du &= \frac & v &= x \end Then :\int \arcsin(x) \, dx = x \arcsin(x) - \int \frac \, dx, which by the simple substitution w=1-x^2,\ dw = -2x\,dx yields the final result: :\int \arcsin(x) \, dx = x \arcsin(x) + \sqrt + C


Extension to the complex plane

Since the inverse trigonometric functions are
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and branch points. One possible way of defining the extension is: :\arctan(z) = \int_0^z \frac \quad z \neq -i, +i where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the branch cut between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For ''z'' not on a branch cut, a straight line path from 0 to ''z'' is such a path. For ''z'' on a branch cut, the path must approach from for the upper branch cut and from for the lower branch cut. The arcsine function may then be defined as: :\arcsin(z) = \arctan\left(\frac\right) \quad z \neq -1, +1 where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between −1 and +1 is the branch cut between the principal sheet of arcsin and other sheets; :\arccos(z) = \frac - \arcsin(z) \quad z \neq -1, +1 which has the same cut as arcsin; :\arccot(z) = \frac - \arctan(z) \quad z \neq -i, i which has the same cut as arctan; :\arcsec(z) = \arccos\left(\frac\right) \quad z \neq -1, 0, +1 where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets; :\arccsc(z) = \arcsin\left(\frac\right) \quad z \neq -1, 0, +1 which has the same cut as arcsec.


Logarithmic forms

These functions may also be expressed using complex logarithms. This extends their domains to the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts. :\begin \arcsin(z) &= -i \ln \left( \sqrt + iz \right) = i \ln \left( \sqrt - iz \right) &= \arccsc\left(\frac\right) \\ 0pt\arccos(z) &= -i \ln \left( i \sqrt + z \right) = \frac - \arcsin(z) &= \arcsec\left(\frac\right) \\ 0pt\arctan(z) &= -\frac\ln \left(\frac\right) = -\frac\ln \left(\frac\right) &= \arccot\left(\frac\right) \\ 0pt\arccot(z) &= -\frac\ln\left( \frac \right) = -\frac\ln\left( \frac \right) &= \arctan\left(\frac\right) \\ 0pt\arcsec(z) &= -i \ln \left( i \sqrt + \frac \right) = \frac - \arccsc(z) &= \arccos\left(\frac\right) \\ 0pt\arccsc(z) &= -i \ln \left( \sqrt + \frac \right) = i \ln \left( \sqrt - \frac \right) &= \arcsin\left(\frac\right) \end


Generalization

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for ...
to form a right triangle in the complex plane. Algebraically, this gives us: :ce^ = c\cos(\theta) + ic\sin(\theta) or :ce^ = a + ib where a is the adjacent side, b is the opposite side, and c is the hypotenuse. From here, we can solve for \theta. :\begin e^ & = a + ib \\ \ln c + i\theta & = \ln(a + ib) \\ \theta & = \operatorname\left( \ln(a + ib) \right) \end or :\theta = -i\ln\left(\frac\right) Simply taking the imaginary part works for any real-valued a and b, but if a or b is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of \ln(a+bi) also removes c from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input z, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the
Pythagorean Theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
relation :a^2 + b^2 = c^2 The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for \theta that result from plugging the values into the equations \theta = -i\ln\left(\tfrac\right) above and simplifying. :\begin & a & & b & & c && -i\ln\left(\frac\right) && \theta && \theta_\\ \arcsin(z)\ \ & \sqrt & & z & & 1 & & -i\ln\left( \frac \right) && = -i\ln\left( \sqrt + iz \right) && \operatorname\left(\ln\left( \sqrt + iz \right)\right) \\ \arccos(z)\ \ & z & & \sqrt & & 1 & & -i\ln\left( \frac \right) && = -i\ln\left( z + \sqrt \right) && \operatorname\left(\ln\left( z + \sqrt \right)\right) \\ \arctan(z)\ \ & 1 & & z & & \sqrt & & -i\ln\left( \frac \right) && = -\frac\ln\left( \frac \right) && \operatorname\left(\ln\left( 1 + iz \right)\right) \\ \arccot(z)\ \ & z & & 1 & & \sqrt & & -i\ln\left( \frac \right) && = -\frac\ln\left( \frac \right) && \operatorname\left(\ln\left( z + i \right)\right) \\ \arcsec(z)\ \ & 1 & & \sqrt & & z & & -i\ln\left( \frac \right) && = -i\ln\left( \frac + \sqrt \right) && \operatorname\left(\ln\left( \frac + \sqrt \right)\right) \\ \arccsc(z)\ \ & \sqrt & & 1 & & z & & -i\ln\left( \frac \right) && = -i\ln\left( \sqrt + \frac \right) && \operatorname\left(\ln\left( \sqrt + \frac \right)\right) \\ \end The particular form of the simplified expression can cause the output to differ from the usual principal branch of each of the inverse trig functions. The formulations given will output the usual principal branch when using the \operatorname\left( \ln z \right) \in (-\pi,\pi] and \operatorname\left(\sqrt\right) \ge 0 principal branch for every function except arccotangent in the \theta column. Arccotangent in the \theta column will output on its usual principal branch by using the \operatorname\left( \ln z \right) \in [0,2\pi) and \operatorname\left(\sqrt\right) \ge 0 convention. In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued z, the definitions allow for hyperbolic angles as outputs and can be used to further define the inverse hyperbolic functions. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.


Example proof

:\begin \sin(\phi) &= z \\ \phi &= \arcsin(z) \end Using the exponential definition of sine, and letting \xi = e^, :\begin z &= \frac \\ 0mu2iz &= \xi - \frac \\ mu0 &= \xi^2 - 2i z \xi - 1 \\ mu\xi &= iz \pm \sqrt \\ mu\phi &= -i \ln \left(iz \pm \sqrt\right) \end (the positive branch is chosen) :\phi= \arcsin(z) = -i \ln \left(iz + \sqrt \right)


Applications


Finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a
right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...
when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that :\theta = \arcsin \left( \frac \right) = \arccos \left( \frac \right) . Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the
Pythagorean Theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
: a^2+b^2=h^2 where h is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed. :\theta = \arctan \left( \frac \right) \, . For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle ''θ'' with the horizontal, where ''θ'' may be computed as follows: :\theta = \arctan \left( \frac \right) = \arctan \left( \frac \right) = \arctan \left( \frac \right) \approx 21.8^ \, .


In computer science and engineering


Two-argument variant of arctangent

The two-argument function computes the arctangent of given and , but with a range of . In other words, is the angle between the positive -axis of a plane and the point on it, with positive sign for counter-clockwise angles (upper half-plane, ), and negative sign for clockwise angles (lower half-plane, ). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering. In terms of the standard arctan function, that is with range of , it can be expressed as follows: \operatorname(y, x) = \begin \arctan\left(\frac y x\right) & \quad x > 0 \\ \arctan\left(\frac y x\right) + \pi & \quad y \ge 0,\; x < 0 \\ \arctan\left(\frac y x\right) - \pi & \quad y < 0,\; x < 0 \\ \frac & \quad y > 0,\; x = 0 \\ -\frac & \quad y < 0,\; x = 0 \\ \text & \quad y = 0,\; x = 0 \end It also equals the principal value of the argument of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. This limited version of the function above may also be defined using the tangent half-angle formulae as follows: \operatorname(y, x) = 2\arctan\left(\frac\right) provided that either or . However this fails if given and so the expression is unsuitable for computational use. The above argument order (, ) seems to be the most common, and in particular is used in
ISO standard The International Organization for Standardization (ISO ; ; ) is an independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. Me ...
s such as the
C programming language C (''pronounced'' '' – like the letter c'') is a general-purpose programming language. It was created in the 1970s by Dennis Ritchie and remains very widely used and influential. By design, C's features cleanly reflect the capabilities of ...
, but a few authors may use the opposite convention (, ) so some caution is warranted.


Arctangent function with location parameter

In many applicationswhen a time varying angle crossing \pm\pi/2 should be mapped by a smooth line instead of a saw toothed one (robotics, astronomy, angular movement in general) the solution y of the equation x=\tan(y) is to come as close as possible to a given value -\infty < \eta < \infty. The adequate solution is produced by the parameter modified arctangent function : y = \arctan_\eta(x) := \arctan(x) + \pi \, \operatorname\left(\frac \right)\, . The function \operatorname rounds to the nearest integer.


Numerical accuracy

For angles near 0 and , arccosine is
ill-conditioned In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
, and similarly with arcsine for angles near −/2 and /2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods.


See also

* Arcsine distribution * Inverse exsecant * Inverse versine *
Inverse hyperbolic functions In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangen ...
* List of integrals of inverse trigonometric functions *
List of trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
*
Trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
* Trigonometric functions of matrices


Notes


References

*


External links

* {{DEFAULTSORT:Inverse Trigonometric Functions Trigonometry Elementary special functions Mathematical relations Ratios Dimensionless numbers