DERIVATIVES OF INVERSE TRIGONOMETRIC FUNCTIONS. The derivative of y = arcsin x. The derivative of y = arccos x. The derivative of y = arctan x. The derivative of y = arccot x. The derivative of y = arcsec x. The derivative of y = arccsc x. I T IS NOT NECESSARY to memorize the derivatives of this Lesson. Rather, the student should know now to derive them. There are several notations used for the inverse trigonometric functions. The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. (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.
Trigonometry |
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Laws and theorems |
Calculus |
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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, 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.
- 2Basic properties
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3In calculus
- 3.3Infinite series
- 3.4Indefinite integrals of inverse trigonometric functions
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4Extension to complex plane
- 4.1Logarithmic forms
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5Applications
- 5.1General solutions
- 5.2In computer science and engineering
Notation[edit]
There are several notations used for the inverse trigonometric functions.
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.[10] In computer programming languages the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan.[citation needed]
The notations sinâ1(x), cosâ1(x), tanâ1(x), etc., as introduced by John Herschel in 1813,[11][12] are often used as well in English-language[6] sources, and this convention complies with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own nameâfor example, (cos(x))â1 = sec(x). Nevertheless, certain authors advise against using it for its ambiguity.[6][13] Another convention used by a few authors is to use a majuscule (capital/upper-case) first letter along with a â1 superscript: Sinâ1(x), Cosâ1(x), Tanâ1(x), etc.[14] This potentially avoids confusion with the multiplicative inverse, which should be represented by sinâ1(x), cosâ1(x), etc.
Since 2009, the ISO 80000-2 standard has specified solely the 'arc' prefix for the inverse functions.
Basic properties[edit]
Principal values[edit]
Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. Therefore, the ranges of the inverse functions are proper 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 = âx 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 â1 ⤠x ⤠1, there are multiple (in fact, countably infinitely many) numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin(Ï) = 0, sin(2Ï) = 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) | â1 ⤠x ⤠1 | âÏ/2 ⤠y ⤠Ï/2 | â90° ⤠y ⤠90° |
arccosine | y = arccos(x) | x = cos(y) | â1 ⤠x ⤠1 | 0 ⤠y â¤ Ï | 0° ⤠y ⤠180° |
arctangent | y = arctan(x) | x = tan(y) | all real numbers | âÏ/2 < y < Ï/2 | â90° < y < 90° |
arccotangent | y = arccot(x) | x = cot(y) | all real numbers | 0 < y < Ï | 0° < y < 180° |
arcsecant | y = arcsec(x) | x = sec(y) | x ⤠â1 or 1 ⤠x | 0 ⤠y < Ï/2 or Ï/2 < y â¤ Ï | 0° ⤠y < 90° or 90° < y ⤠180° |
arccosecant | y = arccsc(x) | x = csc(y) | x ⤠â1 or 1 ⤠x | âÏ/2 ⤠y < 0 or 0 < y ⤠Ï/2 | â90° ⤠y < 0° or 0° < y ⤠90° |
(Note: Some authors define the range of arcsecant to be ( 0 ⤠y < Ï/2 or Ï â¤ y < 3Ï/2 ), because the tangent function is nonnegative on this domain. This makes some computations more consistent. For example, using this range, tan(arcsec(x)) = âx2 â 1, whereas with the range ( 0 ⤠y < Ï/2 or Ï/2 < y â¤ Ï ), we would have to write tan(arcsec(x)) = ±âx2 â 1, since tangent is nonnegative on 0 ⤠y < Ï/2 but nonpositive on Ï/2 < y ⤠Ï. For a similar reason, the same authors define the range of arccosecant to be âÏ < y ⤠âÏ/2 or 0 < y ⤠Ï/2.)
If x is allowed to be a complex number, then the range of y applies only to its real part.
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 (any real number between 0 and 1), then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purely algebraic derivations are longer.[citation needed]
θ{displaystyle theta } | sinâ¡(θ){displaystyle sin(theta )} | cosâ¡(θ){displaystyle cos(theta )} | tanâ¡(θ){displaystyle tan(theta )} | Diagram |
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arcsinâ¡(x){displaystyle arcsin(x)} | sinâ¡(arcsinâ¡(x))=x{displaystyle sin(arcsin(x))=x} | cosâ¡(arcsinâ¡(x))=1âx2{displaystyle cos(arcsin(x))={sqrt {1-x^{2}}}} | tanâ¡(arcsinâ¡(x))=x1âx2{displaystyle tan(arcsin(x))={frac {x}{sqrt {1-x^{2}}}}} | |
arccosâ¡(x){displaystyle arccos(x)} | sinâ¡(arccosâ¡(x))=1âx2{displaystyle sin(arccos(x))={sqrt {1-x^{2}}}} | cosâ¡(arccosâ¡(x))=x{displaystyle cos(arccos(x))=x} | tanâ¡(arccosâ¡(x))=1âx2x{displaystyle tan(arccos(x))={frac {sqrt {1-x^{2}}}{x}}} | |
arctanâ¡(x){displaystyle arctan(x)} | sinâ¡(arctanâ¡(x))=x1+x2{displaystyle sin(arctan(x))={frac {x}{sqrt {1+x^{2}}}}} | cosâ¡(arctanâ¡(x))=11+x2{displaystyle cos(arctan(x))={frac {1}{sqrt {1+x^{2}}}}} | tanâ¡(arctanâ¡(x))=x{displaystyle tan(arctan(x))=x} | |
arccscâ¡(x){displaystyle operatorname {arccsc}(x)} | sinâ¡(arccscâ¡(x))=1x{displaystyle sin(operatorname {arccsc}(x))={frac {1}{x}}} | cosâ¡(arccscâ¡(x))=x2â1x{displaystyle cos(operatorname {arccsc}(x))={frac {sqrt {x^{2}-1}}{x}}} | tanâ¡(arccscâ¡(x))=1x2â1{displaystyle tan(operatorname {arccsc} (x))={frac {1}{sqrt {x^{2}-1}}}} | |
arcsecâ¡(x){displaystyle operatorname {arcsec}(x)} | sinâ¡(arcsecâ¡(x))=x2â1x{displaystyle sin(operatorname {arcsec} (x))={frac {sqrt {x^{2}-1}}{x}}} | cosâ¡(arcsecâ¡(x))=1x{displaystyle cos(operatorname {arcsec}(x))={frac {1}{x}}} | tanâ¡(arcsecâ¡(x))=x2â1{displaystyle tan(operatorname {arcsec}(x))={sqrt {x^{2}-1}}} | |
arccotâ¡(x){displaystyle operatorname {arccot} (x)} | sinâ¡(arccotâ¡(x))=11+x2{displaystyle sin(operatorname {arccot} (x))={frac {1}{sqrt {1+x^{2}}}}} | cosâ¡(arccotâ¡(x))=x1+x2{displaystyle cos(operatorname {arccot}(x))={frac {x}{sqrt {1+x^{2}}}}} | tanâ¡(arccotâ¡(x))=1x{displaystyle tan(operatorname {arccot} (x))={frac {1}{x}}} |
Relationships among the inverse trigonometric functions[edit]
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:
- arccosâ¡(x)=Ï2âarcsinâ¡(x)arccotâ¡(x)=Ï2âarctanâ¡(x)arccscâ¡(x)=Ï2âarcsecâ¡(x){displaystyle {begin{aligned}arccos(x)&={frac {pi }{2}}-arcsin(x)[0.5em]operatorname {arccot} (x)&={frac {pi }{2}}-arctan(x)[0.5em]operatorname {arccsc} (x)&={frac {pi }{2}}-operatorname {arcsec} (x)end{aligned}}}
Negative arguments:
- arcsinâ¡(âx)=âarcsinâ¡(x)arccosâ¡(âx)=Ïâarccosâ¡(x)arctanâ¡(âx)=âarctanâ¡(x)arccotâ¡(âx)=Ïâarccotâ¡(x)arcsecâ¡(âx)=Ïâarcsecâ¡(x)arccscâ¡(âx)=âarccscâ¡(x){displaystyle {begin{aligned}arcsin(-x)&=-arcsin(x)arccos(-x)&=pi -arccos(x)arctan(-x)&=-arctan(x)operatorname {arccot}(-x)&=pi -operatorname {arccot}(x)operatorname {arcsec}(-x)&=pi -operatorname {arcsec}(x)operatorname {arccsc}(-x)&=-operatorname {arccsc}(x)end{aligned}}}
Reciprocal arguments:
- arccosâ¡(1x)=arcsecâ¡(x)arcsinâ¡(1x)=arccscâ¡(x)arctanâ¡(1x)=Ï2âarctanâ¡(x)=arccotâ¡(x), if x>0arctanâ¡(1x)=âÏ2âarctanâ¡(x)=arccotâ¡(x)âÏ, if x<0arccotâ¡(1x)=Ï2âarccotâ¡(x)=arctanâ¡(x), if x>0arccotâ¡(1x)=3Ï2âarccotâ¡(x)=Ï+arctanâ¡(x), if x<0arcsecâ¡(1x)=arccosâ¡(x)arccscâ¡(1x)=arcsinâ¡(x){displaystyle {begin{aligned}arccos left({frac {1}{x}}right)&=operatorname {arcsec}(x)[0.3em]arcsin left({frac {1}{x}}right)&=operatorname {arccsc}(x)[0.3em]arctan left({frac {1}{x}}right)&={frac {pi }{2}}-arctan(x)=operatorname {arccot}(x),{text{ if }}x>0[0.3em]arctan left({frac {1}{x}}right)&=-{frac {pi }{2}}-arctan(x)=operatorname {arccot}(x)-pi ,{text{ if }}x<0[0.3em]operatorname {arccot} left({frac {1}{x}}right)&={frac {pi }{2}}-operatorname {arccot}(x)=arctan(x),{text{ if }}x>0[0.3em]operatorname {arccot} left({frac {1}{x}}right)&={frac {3pi }{2}}-operatorname {arccot}(x)=pi +arctan(x),{text{ if }}x<0[0.3em]operatorname {arcsec} left({frac {1}{x}}right)&=arccos(x)[0.3em]operatorname {arccsc} left({frac {1}{x}}right)&=arcsin(x)end{aligned}}}
Useful identities if one only has a fragment of a sine table:
- arccosâ¡(x)=arcsinâ¡(1âx2), if 0â¤xâ¤1arccosâ¡(x)=12arccosâ¡(2x2â1), if 0â¤xâ¤1arcsinâ¡(x)=12arccosâ¡(1â2x2), if 0â¤xâ¤1arcsinâ¡(x)=arctanâ¡(x1âx2)arctanâ¡(x)=arcsinâ¡(x1+x2){displaystyle {begin{aligned}arccos(x)&=arcsin left({sqrt {1-x^{2}}}right),{text{ if }}0leq xleq 1arccos(x)&={frac {1}{2}}arccos left(2x^{2}-1right),{text{ if }}0leq xleq 1arcsin(x)&={frac {1}{2}}arccos left(1-2x^{2}right),{text{ if }}0leq xleq 1arcsin(x)&=arctan left({frac {x}{sqrt {1-x^{2}}}}right)arctan(x)&=arcsin left({frac {x}{sqrt {1+x^{2}}}}right)end{aligned}}}
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).
From the half-angle formula, tanâ¡(θ2)=sinâ¡(θ)1+cosâ¡(θ){displaystyle tan left({tfrac {theta }{2}}right)={tfrac {sin(theta )}{1+cos(theta )}}}, we get:
- arcsinâ¡(x)=2arctanâ¡(x1+1âx2)arccosâ¡(x)=2arctanâ¡(1âx21+x), if â1<xâ¤+1arctanâ¡(x)=2arctanâ¡(x1+1+x2){displaystyle {begin{aligned}arcsin(x)&=2arctan left({frac {x}{1+{sqrt {1-x^{2}}}}}right)[0.5em]arccos(x)&=2arctan left({frac {sqrt {1-x^{2}}}{1+x}}right),{text{ if }}-1<xleq +1[0.5em]arctan(x)&=2arctan left({frac {x}{1+{sqrt {1+x^{2}}}}}right)end{aligned}}}
Arctangent addition formula[edit]
- arctanâ¡(u)±arctanâ¡(v)=arctanâ¡(u±v1âuv)(modÏ),uvâ 1.{displaystyle arctan(u)pm arctan(v)=arctan left({frac {upm v}{1mp uv}}right){pmod {pi }},quad uvneq 1,.}
This is derived from the tangent addition formula
- tanâ¡(α±β)=tanâ¡(α)±tanâ¡(β)1âtanâ¡(α)tanâ¡(β),{displaystyle tan(alpha pm beta )={frac {tan(alpha )pm tan(beta )}{1mp tan(alpha )tan(beta )}},}
by letting
- α=arctanâ¡(u),β=arctanâ¡(v).{displaystyle alpha =arctan(u),quad beta =arctan(v),.}
In calculus[edit]
Derivatives of inverse trigonometric functions[edit]
The derivatives for complex values of z are as follows:
- ddzarcsinâ¡(z)=11âz2;zâ â1,+1ddzarccosâ¡(z)=â11âz2;zâ â1,+1ddzarctanâ¡(z)=11+z2;zâ âi,+iddzarccotâ¡(z)=â11+z2;zâ âi,+iddzarcsecâ¡(z)=1z21â1z2;zâ â1,0,+1ddzarccscâ¡(z)=â1z21â1z2;zâ â1,0,+1{displaystyle {begin{aligned}{frac {d}{dz}}arcsin(z)&{}={frac {1}{sqrt {1-z^{2}}}};;&z&{}neq -1,+1{frac {d}{dz}}arccos(z)&{}=-{frac {1}{sqrt {1-z^{2}}}};;&z&{}neq -1,+1{frac {d}{dz}}arctan(z)&{}={frac {1}{1+z^{2}}};;&z&{}neq -i,+i{frac {d}{dz}}operatorname {arccot}(z)&{}=-{frac {1}{1+z^{2}}};;&z&{}neq -i,+i{frac {d}{dz}}operatorname {arcsec}(z)&{}={frac {1}{z^{2}{sqrt {1-{frac {1}{z^{2}}}}}}};;&z&{}neq -1,0,+1{frac {d}{dz}}operatorname {arccsc}(z)&{}=-{frac {1}{z^{2}{sqrt {1-{frac {1}{z^{2}}}}}}};;&z&{}neq -1,0,+1end{aligned}}}
Only for real values of x:
- ddxarcsecâ¡(x)=1|x|x2â1;|x|>1ddxarccscâ¡(x)=â1|x|x2â1;|x|>1{displaystyle {begin{aligned}{frac {d}{dx}}operatorname {arcsec}(x)&{}={frac {1}{|x|{sqrt {x^{2}-1}}}};;&|x|>1{frac {d}{dx}}operatorname {arccsc}(x)&{}=-{frac {1}{|x|{sqrt {x^{2}-1}}}};;&|x|>1end{aligned}}}
For a sample derivation: if θ=arcsinâ¡(x){displaystyle theta =arcsin(x)}, we get:
- darcsinâ¡(x)dx=dθdsinâ¡(θ)=dθcosâ¡(θ)dθ=1cosâ¡(θ)=11âsin2â¡(θ)=11âx2{displaystyle {frac {darcsin(x)}{dx}}={frac {dtheta }{dsin(theta )}}={frac {dtheta }{cos(theta )dtheta }}={frac {1}{cos(theta )}}={frac {1}{sqrt {1-sin ^{2}(theta )}}}={frac {1}{sqrt {1-x^{2}}}}}
Expression as definite integrals[edit]
Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:
- arcsinâ¡(x)=â«0x11âz2dz,|x|â¤1arccosâ¡(x)=â«x111âz2dz,|x|â¤1arctanâ¡(x)=â«0x1z2+1dz,arccotâ¡(x)=â«xâ1z2+1dz,arcsecâ¡(x)=â«1x1zz2â1dz=Ï+â«xâ11zz2â1dz,xâ¥1arccscâ¡(x)=â«xâ1zz2â1dz=â«ââx1zz2â1dz,xâ¥1{displaystyle {begin{aligned}arcsin(x)&{}=int _{0}^{x}{frac {1}{sqrt {1-z^{2}}}},dz;,&|x|&{}leq 1arccos(x)&{}=int _{x}^{1}{frac {1}{sqrt {1-z^{2}}}},dz;,&|x|&{}leq 1arctan(x)&{}=int _{0}^{x}{frac {1}{z^{2}+1}},dz;,operatorname {arccot}(x)&{}=int _{x}^{infty }{frac {1}{z^{2}+1}},dz;,operatorname {arcsec}(x)&{}=int _{1}^{x}{frac {1}{z{sqrt {z^{2}-1}}}},dz=pi +int _{x}^{-1}{frac {1}{z{sqrt {z^{2}-1}}}},dz;,&x&{}geq 1operatorname {arccsc}(x)&{}=int _{x}^{infty }{frac {1}{z{sqrt {z^{2}-1}}}},dz=int _{-infty }^{x}{frac {1}{z{sqrt {z^{2}-1}}}},dz;,&x&{}geq 1end{aligned}}}
When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.
Infinite series[edit]
Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows. For arcsine, the series can be derived by expanding its derivative, 11âz2{displaystyle {frac {1}{sqrt {1-z^{2}}}}}, 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{displaystyle {frac {1}{1+z^{2}}}} in a geometric series and applying the integral definition above (see Leibniz series).
- arcsinâ¡(z)=z+(12)z33+(1â 32â 4)z55+(1â 3â 52â 4â 6)z77+â¯=ân=0â(2nâ1)!!(2n)!!â z2n+12n+1=ân=0â(2n)!z2n+1(2n+1)(2nn!)2;|z|â¤1{displaystyle {begin{aligned}arcsin(z)&=z+left({frac {1}{2}}right){frac {z^{3}}{3}}+left({frac {1cdot 3}{2cdot 4}}right){frac {z^{5}}{5}}+left({frac {1cdot 3cdot 5}{2cdot 4cdot 6}}right){frac {z^{7}}{7}}+cdots [5pt]&=sum _{n=0}^{infty }{frac {(2n-1)!!}{(2n)!!}}cdot {frac {z^{2n+1}}{2n+1}}[5pt]&=sum _{n=0}^{infty }{frac {(2n)!,z^{2n+1}}{(2n+1)left(2^{n}n!right)^{2}}},;qquad |z|leq 1end{aligned}}}
- arctanâ¡(z)=zâz33+z55âz77+â¯=ân=0â(â1)nz2n+12n+1;|z|â¤1zâ i,âi{displaystyle arctan(z)=z-{frac {z^{3}}{3}}+{frac {z^{5}}{5}}-{frac {z^{7}}{7}}+cdots =sum _{n=0}^{infty }{frac {(-1)^{n}z^{2n+1}}{2n+1}},;qquad |z|leq 1qquad zneq i,-i}
Series for the other inverse trigonometric functions can be given in terms of these according to the relationships given above. For example, arccosâ¡(x)=Ï/2âarcsinâ¡(x){displaystyle arccos(x)=pi /2-arcsin(x)}, arccscâ¡(x)=arcsinâ¡(1/x){displaystyle operatorname {arccsc} (x)=arcsin(1/x)}, and so on. Another series is given by:[15]
- 2(arcsinâ¡(x2))2=ân=1âx2nn2(2nn){displaystyle 2left(arcsin left({frac {x}{2}}right)right)^{2}=sum _{n=1}^{infty }{frac {x^{2n}}{n^{2}{binom {2n}{n}}}}}
Leonhard Euler found a series for the arctangent that converges more quickly than its Taylor series:
- arctanâ¡(z)=z1+z2ân=0ââk=1n2kz2(2k+1)(1+z2).{displaystyle arctan(z)={frac {z}{1+z^{2}}}sum _{n=0}^{infty }prod _{k=1}^{n}{frac {2kz^{2}}{(2k+1)(1+z^{2})}}.}[16]
(The term in the sum for n = 0 is the empty product, so is 1.)
Alternatively, this can be expressed as
- arctanâ¡(z)=ân=0â22n(n!)2(2n+1)!z2n+1(1+z2)n+1.{displaystyle arctan(z)=sum _{n=0}^{infty }{frac {2^{2n}(n!)^{2}}{(2n+1)!}};{frac {z^{2n+1}}{(1+z^{2})^{n+1}}}.}
Another series for the arctangent function is given by
- arctanâ¡(z)=iân=1â12nâ1(1(1+2i/z)2nâ1â1(1â2i/z)2nâ1),{displaystyle arctan(z)=isum _{n=1}^{infty }{frac {1}{2n-1}}left({frac {1}{(1+2i/z)^{2n-1}}}-{frac {1}{(1-2i/z)^{2n-1}}}right),}
where i=â1{displaystyle i={sqrt {-1}}} is the imaginary unit.[citation needed]
Continued fractions for arctangent[edit]
Two alternatives to the power series for arctangent are these generalized continued fractions:
- arctanâ¡(z)=z1+(1z)23â1z2+(3z)25â3z2+(5z)27â5z2+(7z)29â7z2+â±=z1+(1z)23+(2z)25+(3z)27+(4z)29+â±{displaystyle arctan(z)={frac {z}{1+{cfrac {(1z)^{2}}{3-1z^{2}+{cfrac {(3z)^{2}}{5-3z^{2}+{cfrac {(5z)^{2}}{7-5z^{2}+{cfrac {(7z)^{2}}{9-7z^{2}+ddots }}}}}}}}}}={frac {z}{1+{cfrac {(1z)^{2}}{3+{cfrac {(2z)^{2}}{5+{cfrac {(3z)^{2}}{7+{cfrac {(4z)^{2}}{9+ddots }}}}}}}}}}}
The second of these is valid in the cut complex plane. There are two cuts, from âi 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 â1 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[edit]
For real and complex values of z:
- â«arcsinâ¡(z)dz=zarcsinâ¡(z)+1âz2+Câ«arccosâ¡(z)dz=zarccosâ¡(z)â1âz2+Câ«arctanâ¡(z)dz=zarctanâ¡(z)â12lnâ¡(1+z2)+Câ«arccotâ¡(z)dz=zarccotâ¡(z)+12lnâ¡(1+z2)+Câ«arcsecâ¡(z)dz=zarcsecâ¡(z)âlnâ¡[z(1+z2â1z2)]+Câ«arccscâ¡(z)dz=zarccscâ¡(z)+lnâ¡[z(1+z2â1z2)]+C{displaystyle {begin{aligned}int arcsin(z),dz&{}=z,arcsin(z)+{sqrt {1-z^{2}}}+Cint arccos(z),dz&{}=z,arccos(z)-{sqrt {1-z^{2}}}+Cint arctan(z),dz&{}=z,arctan(z)-{frac {1}{2}}ln left(1+z^{2}right)+Cint operatorname {arccot}(z),dz&{}=z,operatorname {arccot}(z)+{frac {1}{2}}ln left(1+z^{2}right)+Cint operatorname {arcsec}(z),dz&{}=z,operatorname {arcsec}(z)-ln left[zleft(1+{sqrt {frac {z^{2}-1}{z^{2}}}}right)right]+Cint operatorname {arccsc}(z),dz&{}=z,operatorname {arccsc}(z)+ln left[zleft(1+{sqrt {frac {z^{2}-1}{z^{2}}}}right)right]+Cend{aligned}}}
For real x ⥠1:
- â«arcsecâ¡(x)dx=xarcsecâ¡(x)âlnâ¡(x+x2â1)+Câ«arccscâ¡(x)dx=xarccscâ¡(x)+lnâ¡(x+x2â1)+C{displaystyle {begin{aligned}int operatorname {arcsec}(x),dx&{}=x,operatorname {arcsec}(x)-ln left(x+{sqrt {x^{2}-1}}right)+Cint operatorname {arccsc}(x),dx&{}=x,operatorname {arccsc}(x)+ln left(x+{sqrt {x^{2}-1}}right)+Cend{aligned}}}
For all real x not between -1 and 1:
- â«arcsecâ¡(x)dx=xarcsecâ¡(x)âsgnâ¡(x)lnâ¡(|x+x2â1|)+Câ«arccscâ¡(x)dx=xarccscâ¡(x)+sgnâ¡(x)lnâ¡(|x+x2â1|)+C{displaystyle {begin{aligned}int operatorname {arcsec}(x),dx&{}=x,operatorname {arcsec}(x)-operatorname {sgn}(x)ln left(left|x+{sqrt {x^{2}-1}}right|right)+Cint operatorname {arccsc}(x),dx&{}=x,operatorname {arccsc}(x)+operatorname {sgn}(x)ln left(left|x+{sqrt {x^{2}-1}}right|right)+Cend{aligned}}}
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:
- â«arcsecâ¡(x)dx=xarcsecâ¡(x)âarcoshâ¡(|x|)+Câ«arccscâ¡(x)dx=xarccscâ¡(x)+arcoshâ¡(|x|)+C{displaystyle {begin{aligned}int operatorname {arcsec}(x),dx&{}=x,operatorname {arcsec}(x)-operatorname {arcosh} (|x|)+Cint operatorname {arccsc}(x),dx&{}=x,operatorname {arccsc}(x)+operatorname {arcosh} (|x|)+Cend{aligned}}}
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[edit]
Using â«udv=uvââ«vdu{displaystyle int u,dv=uv-int v,du} (i.e. integration by parts), set
- u=arcsinâ¡(x)dv=dxdu=dx1âx2v=x{displaystyle {begin{aligned}u&=arcsin(x)&dv&=dxdu&={frac {dx}{sqrt {1-x^{2}}}}&v&=xend{aligned}}}
Then
- â«arcsinâ¡(x)dx=xarcsinâ¡(x)ââ«x1âx2dx,{displaystyle int arcsin(x),dx=xarcsin(x)-int {frac {x}{sqrt {1-x^{2}}}},dx,}
which by a simple substitution yields the final result:
- â«arcsinâ¡