Inverse trigonometric functions

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Short description: Inverse functions of sin, cos, tan, etc.

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.

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 , 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: tan1(x)={arctan(x)+πkkZ} . 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

The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.

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 y=x could be defined from y2=x, the function y=arcsin(x) is defined so that sin(y)=x. For a given real number x, with 1x1, there are multiple (in fact, countably infinitely many) numbers y such that sin(y)=x; for example, sin(0)=0, but also sin(π)=0, sin(2π)=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.

Name Usual notation Definition Domain of x for real result Range of usual principal value
(radians)
Range of usual principal value
(degrees)
arcsine y=arcsin(x) x = sin(y) 1x1 π2yπ2 90y90
arccosine y=arccos(x) x = cos(y) 1x1 0yπ 0y180
arctangent y=arctan(x) x = tan(y) all real numbers π2<y<π2 90<y<90
arccotangent y=\arccot(x) x = cot(y) all real numbers 0<y<π 0<y<180
arcsecant y=\arcsec(x) x = sec(y) |x|1 0y<π2 or π2<yπ 0y<90 or 90<y180
arccosecant y=\arccsc(x) x = csc(y) |x|1 π2y<0 or 0<yπ2 90y<0 or 0<y90

Note: Some authors define the range of arcsecant to be (0y<π2 or πy<3π2), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan(\arcsec(x))=x21, whereas with the range (0y<π2 or π2<yπ), we would have to write tan(\arcsec(x))=±x21, since tangent is nonnegative on 0y<π2, but nonpositive on π2<yπ. For a similar reason, the same authors define the range of arccosecant to be (π<yπ2 or 0<yπ2).

Domains

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

Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions

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 2π:

  • Sine and cosecant begin their period at 2πkπ2 (where k is an integer), finish it at 2πk+π2, and then reverse themselves over 2πk+π2 to 2πk+3π2.
  • Cosine and secant begin their period at 2πk, finish it at 2πk+π. and then reverse themselves over 2πk+π to 2πk+2π.
  • Tangent begins its period at 2πkπ2, finishes it at 2πk+π2, and then repeats it (forward) over 2πk+π2 to 2πk+3π2.
  • Cotangent begins its period at 2πk, finishes it at 2πk+π, and then repeats it (forward) over 2πk+π to 2πk+2π.

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 θ, r, s, x, and y all lie within appropriate ranges so that the relevant expressions below are well-defined. Note that "for some kZ" is just another way of saying "for some integer k."

The symbol 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 footnote[note 1] for more details and an example illustrating this concept).

Equation if and only if Solution
sinθ=y θ= (1)k arcsin(y) + πk for some kZ
cscθ=r θ= (1)k \arccsc(r) + πk for some kZ
cosθ=x θ= ± arccos(x) + 2 πk for some kZ
secθ=r θ= ± \arcsec(r) + 2 πk for some kZ
tanθ=s θ= arctan(s) + πk for some kZ
cotθ=r θ= \arccot(r) + πk for some kZ

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

Equation if and only if Solution
sinθ=y θ=arcsin(y)+2πh
          or
θ=arcsin(y)+2πh+π
for some hZ
cscθ=r θ=\arccsc(r)+2πh
          or
θ=\arccsc(r)+2πh+π
for some hZ
cosθ=x θ=arccos(x)+2πk
         or
θ=arccos(x)+2πk
for some kZ
secθ=r θ=\arcsec(r)+2πk
         or
θ=\arcsec(r)+2πk
for some kZ

For example, if cosθ=1 then θ=π+2πk=π+2π(1+k) for some kZ. While if sinθ=±1 then θ=π2+πk=π2+π(k+1) for some kZ, where k will be even if sinθ=1 and it will be odd if sinθ=1. The equations secθ=1 and cscθ=±1 have the same solutions as cosθ=1 and sinθ=±1, respectively. In all equations above except for those just solved (i.e. except for sin/cscθ=±1 and cos/secθ=1), the integer k in the solution's formula is uniquely determined by θ (for fixed r,s,x, and y).

With the help of integer parity Parity(h)={0if h is even 1if h is odd  it is possible to write a solution to cosθ=x that doesn't involve the "plus or minus" ± symbol:

cosθ=x if and only if θ=(1)harccos(x)+πh+πParity(h) for some hZ.

And similarly for the secant function,

secθ=r if and only if θ=(1)h\arcsec(r)+πh+πParity(h) for some hZ,

where πh+πParity(h) equals πh when the integer h is even, and equals πh+π when it's odd.

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

The solutions to cosθ=x and secθ=x involve the "plus or minus" symbol ±, whose meaning is now clarified. Only the solution to cosθ=x will be discussed since the discussion for secθ=x is the same. We are given x between 1x1 and we know that there is an angle θ in some interval that satisfies cosθ=x. We want to find this θ. The table above indicates that the solution is θ=±arccosx+2πk for some kZ which is a shorthand way of saying that (at least) one of the following statement is true:

  1. θ=arccosx+2πk for some integer k,
    or
  2. θ=arccosx+2πk for some integer k.

As mentioned above, if arccosx=π (which by definition only happens when x=cosπ=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 θ=π+2πK holds, then the integer k for statement (2) is K+1 (because θ=π+2π(1+K)). However, if x1 then the integer k is unique and completely determined by θ. If arccosx=0 (which by definition only happens when x=cos0=1) then ±arccosx=0 (because +arccosx=+0=0 and arccosx=0=0 so in both cases ±arccosx 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 arccosx=0 and arccosx=π, we now focus on the case where arccosx0 and arccosxπ, So assume this from now on. The solution to cosθ=x is still θ=±arccosx+2πk for some kZ which as before is shorthand for saying that one of statements (1) and (2) is true. However this time, because arccosx0 and 0<arccosx<π, statements (1) and (2) are different and furthermore, exactly one of the two equalities holds (not both). Additional information about θ is needed to determine which one holds. For example, suppose that x=0 and that all that is known about θ is that πθπ (and nothing more is known). Then arccosx=arccos0=π2 and moreover, in this particular case k=0 (for both the + case and the case) and so consequently, θ = ±arccosx+2πk = ±(π2)+2π(0) = ±π2. This means that θ could be either π/2 or π/2. Without additional information it is not possible to determine which of these values θ has. An example of some additional information that could determine the value of θ would be knowing that the angle is above the x-axis (in which case θ=π/2) or alternatively, knowing that it is below the x-axis (in which case θ=π/2).

Equal identical trigonometric functions

Set of all solutions to elementary trigonometric equations

Thus given a single solution θ to an elementary trigonometric equation (sinθ=y is such an equation, for instance, and because sin(arcsiny)=y always holds, θ:=arcsiny is always a solution), the set of all solutions to it are:

If θ solves then Set of all solutions (in terms of θ)
sinθ=y then {φ:sinφ=y}= (θ +2 πZ) (θ π +2πZ)
cscθ=r then {φ:cscφ=r}= (θ +2 πZ) (θ π +2πZ)
cosθ=x then {φ:cosφ=x}= (θ +2 πZ) (θ +2πZ)
secθ=r then {φ:secφ=r}= (θ +2 πZ) (θ +2πZ)
tanθ=s then {φ:tanφ=s}= θ + πZ
cotθ=r then {φ:cotφ=r}= θ + πZ

Transforming equations

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

Transforming equations by shifts and reflections
Argument:       = θ π2±θ π±θ 3π2±θ 2kπ±θ,
(kZ)
sin              = sinθ cosθ sinθ cosθ ±sinθ
csc              = cscθ secθ cscθ secθ ±cscθ
cos              = cosθ sinθ cosθ ±sinθ cosθ
sec              = secθ cscθ secθ ±cscθ secθ
tan              = tanθ cotθ ±tanθ cotθ ±tanθ
cot              = cotθ tanθ ±cotθ tanθ ±cotθ

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

sinθ=sin(θ)=sin(π+θ)=sin(πθ)=cos(π2+θ)=cos(π2θ)=cos(π2θ)=cos(π2+θ)=cos(3π2θ)=cos(3π2+θ)cosθ=cos(θ)=cos(π+θ)=cos(πθ)=sin(π2+θ)=sin(π2θ)=sin(π2θ)=sin(π2+θ)=sin(3π2θ)=sin(3π2+θ)tanθ=tan(θ)=tan(π+θ)=tan(πθ)=cot(π2+θ)=cot(π2θ)=cot(π2θ)=cot(π2+θ)=cot(3π2θ)=cot(3π2+θ)

where swapping sincsc, swapping cossec, and swapping tancot gives the analogous equations for csc,sec, and cot, respectively.

So for example, by using the equality sin(π2θ)=cosθ, the equation cosθ=x can be transformed into sin(π2θ)=x, which allows for the solution to the equation sinφ=x (where φ:=π2θ) to be used; that solution being: φ=(1)karcsin(x)+πk for some kZ, which becomes: π2θ = (1)karcsin(x)+πk for some kZ where using the fact that (1)k=(1)k and substituting h:=k proves that another solution to cosθ=x is: θ = (1)h+1arcsin(x)+πh+π2 for some hZ. The substitution arcsinx=π2arccosx may be used express the right hand side of the above formula in terms of arccosx instead of arcsinx.

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 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 values and the signum (sgn) operation.

θ sin(θ) cos(θ) tan(θ) Diagram
arcsin(x) sin(arcsin(x))=x cos(arcsin(x))=1x2 tan(arcsin(x))=x1x2 Trigonometric functions and inverse3.svg
arccos(x) sin(arccos(x))=1x2 cos(arccos(x))=x tan(arccos(x))=1x2x Trigonometric functions and inverse.svg
arctan(x) sin(arctan(x))=x1+x2 cos(arctan(x))=11+x2 tan(arctan(x))=x Trigonometric functions and inverse2.svg
\arccot(x) sin(\arccot(x))=11+x2 cos(\arccot(x))=x1+x2 tan(\arccot(x))=1x Trigonometric functions and inverse4.svg
\arcsec(x) sin(\arcsec(x))=x21|x| cos(\arcsec(x))=1x tan(\arcsec(x))=\sgn(x)x21 Trigonometric functions and inverse6.svg
\arccsc(x) sin(\arccsc(x))=1x cos(\arccsc(x))=x21|x| tan(\arccsc(x))=\sgn(x)x21 Trigonometric functions and inverse5.svg

Relationships among the inverse trigonometric functions

The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.
The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.
Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

Complementary angles:

arccos(x)=π2arcsin(x)\arccot(x)=π2arctan(x)\arccsc(x)=π2\arcsec(x)

Negative arguments:

arcsin(x)=arcsin(x)\arccsc(x)=\arccsc(x)arccos(x)=πarccos(x)\arcsec(x)=π\arcsec(x)arctan(x)=arctan(x)\arccot(x)=π\arccot(x)

Reciprocal arguments:

arcsin(1x)=\arccsc(x)\arccsc(1x)=arcsin(x)arccos(1x)=\arcsec(x)\arcsec(1x)=arccos(x)arctan(1x)=\arccot(x)=π2arctan(x), if x>0arctan(1x)=\arccot(x)π=π2arctan(x), if x<0\arccot(1x)=arctan(x)=π2\arccot(x), if x>0\arccot(1x)=arctan(x)+π=3π2\arccot(x), if x<0

The identities above can be used with (and derived from) the fact that sin and csc are reciprocals (i.e. csc=1sin), as are cos and sec, and tan and cot.

Useful identities if one only has a fragment of a sine table:

arcsin(x)=12arccos(12x2), if 0x1arcsin(x)=arctan(x1x2)arccos(x)=12arccos(2x21), if 0x1arccos(x)=arctan(1x2x)arccos(x)=arcsin(1x2), if 0x1 , from which you get arccos(1x21+x2)=arcsin(2x1+x2), if 0x1arcsin(1x2)=π2\sgn(x)arcsin(x)arctan(x)=arcsin(x1+x2)\arccot(x)=arccos(x1+x2)

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(11+x2), if x0.

It is obtained by recognizing that cos(arctan(x))=11+x2=cos(arccos(11+x2)).

From the half-angle formula, tan(θ2)=sin(θ)1+cos(θ), we get:

arcsin(x)=2arctan(x1+1x2)arccos(x)=2arctan(1x21+x), if 1<x1arctan(x)=2arctan(x1+1+x2)

Arctangent addition formula

arctan(u)±arctan(v)=arctan(u±v1uv)(modπ),uv1.

This is derived from the tangent addition formula

tan(α±β)=tan(α)±tan(β)1tan(α)tan(β),

by letting

α=arctan(u),β=arctan(v).

In calculus

Derivatives of inverse trigonometric functions

Main page: Differentiation of trigonometric functions

The derivatives for complex values of z are as follows:

ddzarcsin(z)=11z2;z1,+1ddzarccos(z)=11z2;z1,+1ddzarctan(z)=11+z2;zi,+iddz\arccot(z)=11+z2;zi,+iddz\arcsec(z)=1z211z2;z1,0,+1ddz\arccsc(z)=1z211z2;z1,0,+1

Only for real values of x:

ddx\arcsec(x)=1|x|x21;|x|>1ddx\arccsc(x)=1|x|x21;|x|>1

These formulas can be derived in terms of the derivatives of trigonometric functions. For example, if x=sinθ, then dx/dθ=cosθ=1x2, so

ddxarcsin(x)=dθdx=1dx/dθ=11x2.

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:

arcsin(x)=0x11z2dz,|x|1arccos(x)=x111z2dz,|x|1arctan(x)=0x1z2+1dz,\arccot(x)=x1z2+1dz,\arcsec(x)=1x1zz21dz=π+x11zz21dz,x1\arccsc(x)=x1zz21dz=x1zz21dz,x1

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series

Similar to the sine and cosine functions, the inverse trigonometric functions can also be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, 11z2, as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 11+z2 in a geometric series, and applying the integral definition above (see Leibniz series).

arcsin(z)=z+(12)z33+(1324)z55+(135246)z77+=n=0(2n1)!!(2n)!!z2n+12n+1=n=0(2n)!(2nn!)2z2n+12n+1;|z|1
arctan(z)=zz33+z55z77+=n=0(1)nz2n+12n+1;|z|1zi,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)=π/2arcsin(x), \arccsc(x)=arcsin(1/x), and so on. Another series is given by:[19]

2(arcsin(x2))2=n=1x2nn2(2nn).

Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:

arctan(z)=z1+z2n=0k=1n2kz2(2k+1)(1+z2).[20]

(The term in the sum for n = 0 is the empty product, so is 1.)

Alternatively, this can be expressed as

arctan(z)=n=022n(n!)2(2n+1)!z2n+1(1+z2)n+1.

Another series for the arctangent function is given by

arctan(z)=in=112n1(1(1+2i/z)2n11(12i/z)2n1),

where i=1 is the imaginary unit.[21]

Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

arctan(z)=z1+(1z)231z2+(3z)253z2+(5z)275z2+(7z)297z2+=z1+(1z)23+(2z)25+(3z)27+(4z)29+

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; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Indefinite integrals of inverse trigonometric functions

For real and complex values of z:

arcsin(z)dz=zarcsin(z)+1z2+Carccos(z)dz=zarccos(z)1z2+Carctan(z)dz=zarctan(z)12ln(1+z2)+C\arccot(z)dz=z\arccot(z)+12ln(1+z2)+C\arcsec(z)dz=z\arcsec(z)ln[z(1+z21z2)]+C\arccsc(z)dz=z\arccsc(z)+ln[z(1+z21z2)]+C

For real x ≥ 1:

\arcsec(x)dx=x\arcsec(x)ln(x+x21)+C\arccsc(x)dx=x\arccsc(x)+ln(x+x21)+C

For all real x not between -1 and 1:

\arcsec(x)dx=x\arcsec(x)\sgn(x)ln|x+x21|+C\arccsc(x)dx=x\arccsc(x)+\sgn(x)ln|x+x21|+C

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 derivatives 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 functions:

\arcsec(x)dx=x\arcsec(x)arcosh(|x|)+C\arccsc(x)dx=x\arccsc(x)+arcosh(|x|)+C

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 and the simple derivative forms shown above.

Example

Using udv=uvvdu (i.e. integration by parts), set

u=arcsin(x)dv=dxdu=dx1x2v=x

Then

arcsin(x)dx=xarcsin(x)x1x2dx,

which by the simple substitution w=1x2, dw=2xdx yields the final result:

arcsin(x)dx=xarcsin(x)+1x2+C

Extension to the complex plane

A Riemann surface for the argument of the relation tan z = x. The orange sheet in the middle is the principal sheet representing arctan x. The blue sheet above and green sheet below are displaced by 2π and −2π respectively.

Since the inverse trigonometric functions are analytic functions, 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)=0zdx1+x2zi,+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 Re[x] > 0 for the upper branch cut and from Re[x] < 0 for the lower branch cut.

The arcsine function may then be defined as:

arcsin(z)=arctan(z1z2)z1,+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)=π2arcsin(z)z1,+1

which has the same cut as arcsin;

\arccot(z)=π2arctan(z)zi,i

which has the same cut as arctan;

\arcsec(z)=arccos(1z)z1,0,+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(1z)z1,0,+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 a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

arcsin(z)=iln(1z2+iz)=iln(1z2iz)=\arccsc(1z)arccos(z)=iln(i1z2+z)=π2arcsin(z)=\arcsec(1z)arctan(z)=i2ln(izi+z)=i2ln(1+iz1iz)=\arccot(1z)\arccot(z)=i2ln(z+izi)=i2ln(iz1iz+1)=arctan(1z)\arcsec(z)=iln(i11z2+1z)=π2\arccsc(z)=arccos(1z)\arccsc(z)=iln(11z2+iz)=iln(11z2iz)=arcsin(1z)

Generalization

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using Euler's formula to form a right triangle in the complex plane. Algebraically, this gives us:

ceiθ=ccos(θ)+icsin(θ)

or

ceiθ=a+ib

where a is the adjacent side, b is the opposite side, and c is the hypotenuse. From here, we can solve for θ.

eln(c)+iθ=a+iblnc+iθ=ln(a+ib)θ=Im(ln(a+ib))

or

θ=iln(a+ibc)

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 relation

a2+b2=c2

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for θ that result from plugging the values into the equations θ=iln(a+ibc) above and simplifying.

abciln(a+ibc)θθa,bRarcsin(z)  1z2z1iln(1z2+iz1)=iln(1z2+iz)Im(ln(1z2+iz))arccos(z)  z1z21iln(z+i1z21)=iln(z+z21)Im(ln(z+z21))arctan(z)  1z1+z2iln(1+iz1+z2)=iln(1+iz1+z2)Im(ln(1+iz))\arccot(z)  z1z2+1iln(z+iz2+1)=iln(z+iz2+1)Im(ln(z+i))\arcsec(z)  1z21ziln(1+iz21z)=iln(1z+1z21)Im(ln(1z+1z21))\arccsc(z)  z211ziln(z21+iz)=iln(11z2+iz)Im(ln(11z2+iz))

In order to match the principal branch of the natural log and square root functions to the usual principal branch of the inverse trig functions, the particular form of the simplified formulation matters. The formulations given in the two rightmost columns assume Im(lnz)(π,π] and Re(z)0. To match the principal branch Im(lnz)[0,2π) and Im(z)0 to the usual principal branch of the inverse trig functions, subtract 2π from the result θ when Re(θ)>π.

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. Elementary proofs of the relations may also proceed via expansion to exponential forms of the trigonometric functions.

Example proof

sin(ϕ)=zϕ=arcsin(z)

Using the exponential definition of sine, and letting ξ=eiϕ,

z=eiϕeiϕ2i2iz=ξ1ξ0=ξ22izξ1ξ=iz±1z2ϕ=iln(iz±1z2)

(the positive branch is chosen)

ϕ=arcsin(z)=iln(iz+1z2)
Color wheel graphs of inverse trigonometric functions in the complex plane
Arcsine of z in the complex plane. Arccosine of z in the complex plane. Arctangent of z in the complex plane.
arcsin(z) arccos(z) arctan(z)
Arccosecant of z in the complex plane. Arcsecant of z in the complex plane. Arccotangent of z in the complex plane.
\arccsc(z) \arcsec(z) \arccot(z)

Applications

Finding the angle of a right triangle

A right triangle with sides relative to an angle at the A point.

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine and cosine, it follows that

θ=arcsin(oppositehypotenuse)=arccos(adjacenthypotenuse).

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: a2+b2=h2 where h is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

θ=arctan(oppositeadjacent).

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:

θ=arctan(oppositeadjacent)=arctan(riserun)=arctan(820)21.8.

In computer science and engineering

Two-argument variant of arctangent

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−ππ]. In other words, atan2(yx) is the angle between the positive x-axis of a plane and the point (xy) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). 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 (−π/2, π/2), it can be expressed as follows:

atan2(y,x)={arctan(yx)x>0arctan(yx)+πy0,x<0arctan(yx)πy<0,x<0π2y>0,x=0π2y<0,x=0undefinedy=0,x=0

It also equals the principal value of the argument of the complex number x + iy.

This limited version of the function above may also be defined using the tangent half-angle formulae as follows:

atan2(y,x)=2arctan(yx2+y2+x)

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.

The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at atan2.

Arctangent function with location parameter

In many applications[22] the solution y of the equation x=tan(y) is to come as close as possible to a given value <η<. The adequate solution is produced by the parameter modified arctangent function

y=arctanη(x):=arctan(x)+πrni(ηarctan(x)π).

The function rni rounds to the nearest integer.

Numerical accuracy

For angles near 0 and π, arccosine is ill-conditioned, and similarly with arcsine for angles near −π/2 and π/2. Computer applications thus need to consider the stability of inputs to these functions and the sensitivity of their calculations, or use alternate methods.[23]

See also


Notes

  1. The expression "LHS RHS" indicates that either (a) the left hand side (i.e. LHS) and right hand side (i.e. RHS) are both true, or else (b) the left hand side and right hand side are both false; there is no option (c) (e.g. it is not possible for the LHS statement to be true and also simultaneously for the RHS statement to be false), because otherwise "LHS RHS" would not have been written.
    To clarify, suppose that it is written "LHS 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 θ and s are some given and fixed numbers and if the following is written: tanθ=sθ=arctan(s)+πk for some kZ then LHS is the statement "tanθ=s". Depending on what specific values θ and s have, this LHS statement can either be true or false. For instance, LHS is true if θ=0 and s=0 (because in this case tanθ=tan0=s) but LHS is false if θ=0 and s=2 (because in this case tanθ=tan0=s which is not equal to s=2); more generally, LHS is false if θ=0 and s0. Similarly, RHS is the statement "θ=arctan(s)+πk for some kZ". The RHS statement can also either true or false (as before, whether the RHS statement is true or false depends on what specific values θ and s have). The logical equality symbol means that (a) if the LHS statement is true then the RHS statement is also necessarily true, and moreover (b) if the LHS statement is false then the RHS statement is also necessarily false. Similarly, also means that (c) if the RHS statement is true then the LHS statement is also necessarily true, and moreover (d) if the RHS statement is false then the LHS statement is also necessarily false.

References

  1. "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods 155 (3): 543–546. 1978-10-01. doi:10.1016/0029-554X(78)90541-4. Bibcode1978NucIM.155..543T. 
  2. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. 1994. ISBN 978-155608010-4. 
  3. Preparatory Course in Mathematics (6 ed.). Department of Physics, University of Konstanz. 2005-07-25. https://www.math.uni-konstanz.de/numerik/personen/gubisch/de/teaching/ws0708/vorkurs-skript.pdf. Retrieved 2017-07-26. 
  4. Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (1 ed.). Ventus Publishing ApS / Bookboon. 2010-11-11. ISBN 978-87-7681-702-2. http://netsaver.myds.me/sym/pub/Netsaver%20Library/Mejlbro,%20Leif/Complex%20Functions%20Theory,%20vol.3%20-%20S%20(2365)/Complex%20Functions%20Theory,%20vol.3%20-%20Mejlbro,%20Leif.pdf. Retrieved 2017-07-26. 
  5. Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. 2012. p. 88. ISBN 978-956141314-6. 
  6. Jump up to: 6.0 6.1 6.2 6.3 "Chapter II. The Acute Angle [14 Inverse trigonometric functions"]. written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. January 1909. p. 15. https://archive.org/stream/planetrigonometr00hallrich#page/n30/mode/1up. Retrieved 2017-08-12. "[…] α = arcsin m: It is frequently read "arc-sine m" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […]" 
  7. (in de) Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis. 1 (3rd ed.). Berlin: J. Springer. 1924. 
  8. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. 2004. ISBN 978-0-48643480-3. https://books.google.com/books?id=8KuoxgykfbkC. Retrieved 2017-08-13. 
  9. Triumph der Mathematik. Dover Publications. 1965. p. 69. ISBN 978-0-486-61348-2. 
  10. Weisstein, Eric W.. "Inverse Trigonometric Functions" (in en). https://mathworld.wolfram.com/InverseTrigonometricFunctions.html. 
  11. "Inverse trigonometric functions". The Americana: a universal reference library. 21. 1912. 
  12. Cook, John D. (2021-02-11). "Trig functions across programming languages". https://www.johndcook.com/blog/2021/02/11/trig-across-languages. 
  13. A History of Mathematics (2 ed.). New York, NY: The Macmillan Company. 1919. p. 272. https://archive.org/details/ahistorymathema02cajogoog. 
  14. "On a remarkable Application of Cotes's Theorem". Philosophical Transactions (Royal Society, London) 103 (1): 8. 1813. doi:10.1098/rstl.1813.0005. https://books.google.com/books?id=qpRJAAAAYAAJ&pg=PA8. 
  15. "Inverse trigonometric functions". https://brilliant.org/wiki/inverse-trigonometric-functions/. 
  16. Korn, Grandino Arthur; Korn, Theresa M. (2000). "21.2.-4. Inverse Trigonometric Functions". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulars for reference and review (3 ed.). Mineola, New York, USA: Dover Publications, Inc.. p. 811. ISBN 978-0-486-41147-7. https://archive.org/details/mathematicalhand00korn_849. 
  17. "Differentiation of Trigonometric, Logarithmic and Exponential Functions" (in en-PK). Calculus and Analytic Geometry (1 ed.). Lahore: Punjab Textbook Board. 1999. p. 140. 
  18. Abramowitz & Stegun 1972, p. 73, 4.3.44
  19. Experimentation in Mathematics: Computational Paths to Discovery (1 ed.). Wellesley, MA, USA: A. K. Peters. 2004. p. 51. ISBN 978-1-56881-136-9. https://archive.org/details/experimentationm00borw_656. 
  20. Hwang Chien-Lih (2005), "An elementary derivation of Euler's series for the arctangent function", The Mathematical Gazette 89 (516): 469–470, doi:10.1017/S0025557200178404 
  21. S. M. Abrarov and B. M. Quine (2018), "A formula for pi involving nested radicals", The Ramanujan Journal 46 (3): 657–665, doi:10.1007/s11139-018-9996-8 
  22. when a time varying angle crossing ±π/2 should be mapped by a smooth line instead of a saw toothed one (robotics, astromomy, angular movement in general)[citation needed]
  23. "A non-singular horizontal position representation". The Journal of Navigation (Cambridge University Press) 63 (3): 395–417. 2010. doi:10.1017/S0373463309990415. Bibcode2010JNav...63..395G. http://www.navlab.net/Publications/A_Nonsingular_Horizontal_Position_Representation.pdf. 

External links