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Inverse Sine

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ArcSin
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The inverse sine is the multivalued function sin^(-1)z (Zwillinger 1995, p. 465), also denoted arcsinz (Abramowitz and Stegun 1972, p. 79; Harris and Stocker 1998, p. 307; Jeffrey 2000, p. 124), that is the inverse function of the sine. The variants Arcsinz (e.g., Bronshtein and Semendyayev, 1997, p. 69) and Sin^(-1)z are sometimes used to refer to explicit principal values of the inverse sine, although this distinction is not always made (e.g,. Zwillinger 1995, p. 466). Worse yet, the notation arcsinz is sometimes used for the principal value, with Arcsinz being used for the multivalued function (Abramowitz and Stegun 1972, p. 80). Note that in the notation sin^(-1)z (commonly used in North America and in pocket calculators worldwide), sinz is the sine and the superscript -1 denotes the inverse function, not the multiplicative inverse.

The principal value of the inverse sine is implemented as ArcSin[z] in the Wolfram Language. In the GNU C library, it is implemented as asin(double x).

InverseSineBranchCut

The inverse sine is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at (-infty,-1) and (1,infty). This follows from the definition of sin^(-1)z as

sin^(-1)z=-iln(iz+sqrt(1-z^2)).
(1)

Special values include

sin^(-1)(-1)=-1/2pi
(2)
sin^(-1)0=0
(3)
sin^(-1)1=1/2pi.
(4)

The derivative of sin^(-1)z is

d/(dz)sin^(-1)z=1/(sqrt(1-z^2))
(5)

and its indefinite integral is

intsin^(-1)zdz=sqrt(1-z^2)+zsin^(-1)z+C.
(6)

The inverse sine satisfies

sin^(-1)z=csc^(-1)(1/z)
(7)

for z!=0,

sin^(-1)z=-sin^(-1)(-z)
(8)
=cos^(-1)(-z)-1/2pi
(9)
=1/2pi-cos^(-1)z
(10)

for all complex z,

sin^(-1)x={-1/2pi+sin^(-1)(sqrt(1-x^2)) for x<0; 1/2pi-sin^(-1)(sqrt(1-x^2)) for x>0
(11)
={-1/2pi-cot^(-1)(x/(sqrt(1-x^2))) for x<0; 1/2pi-cot^(-1)(x/(sqrt(1-x^2))) for x>0
(12)
={-cos^(-1)(sqrt(1-x^2)) for -1<x<0; cos^(-1)(sqrt(1-x^2)) for 0<x<1
(13)
={-sec^(-1)(1/(sqrt(1-x^2))) for -1<x<0; sec^(-1)(1/(sqrt(1-x^2))) for 0<x<1,
(14)

and

sin^(-1)x=tan^(-1)(x/(sqrt(1-x^2)))
(15)
=cot^(-1)((sqrt(1-x^2))/x)
(16)

for -1<x<1, where equality at points where the denominators are 0 is understood to mean in the limit as x->+/-1 or x->0, respectively.

The Maclaurin series for the inverse sine with -1<=x<=1 is given by

sin^(-1)x=sum_(n=0)^(infty)((1/2)_n)/((2n+1)n!)x^(2n+1)
(17)
=x+1/6x^3+3/(40)x^5+5/(112)x^7+(35)/(1152)x^9+...
(18)

(OEIS A055786 and A002595), where (x)_n is a Pochhammer symbol.

The inverse sine can be given by the sum

(sin^(-1)x)^2=1/2sum_(n=1)^infty((2x)^(2n))/(n^2(2n; n)),
(19)

where (2n; n) is a binomial coefficient (Borwein et al. 2004, p. 51; Borwein and Chamberland 2005; Bailey et al. 2007, pp. 15-16). Similarly,

[sin^(-1)(1/2x)]^4=3/2sum_(k=1)^(infty)[sum_(m=1)^(k-1)1/(m^2)](x^(2k))/(k^2(2k; k))
(20)
[sin^(-1)(1/2x)]^6=(45)/4sum_(k=1)^(infty)[sum_(m=1)^(k-1)1/(m^2)sum_(n=1)^(m-1)1/(n^2)](x^(2k))/(k^2(2k; k))
(21)
[sin^(-1)(1/2x)]^8=(315)/2sum_(k=1)^(infty)[sum_(m=1)^(k-1)1/(m^2)sum_(n=1)^(m-1)1/(n^2)sum_(p=1)^(n-1)1/(p^2)](x^(2k))/(k^2(2k; k))
(22)

(Bailey et al. 2007, pp. 16 and 282; Borwein and Chamberland 2007). Ramanujan gave the cases (sin^(-1)x)^n for n=1, 2, 3, and 4 (Berndt 1985, pp. 262-263), and the general cases are given in terms of multiple sums by Bailey et al. (2006, pp. 15-16 and 282) and Borwein and Chamberland (2007).

The inverse sine has continued fraction

sin^(-1)z=(zsqrt(1-z^2))/(1-(1·2z^2)/(3-(1·2z^2)/(5-(3·4z^2)/(7-(3·4z^2)/(9-(5·6z^2)/(11-...))))))
(23)

(Wall 1948, p. 345).

See also

Inverse Cosecant, Inverse Cosine, Inverse Cotangent, Inverse Secant, Inverse Tangent, Inverse Trigonometric Functions, Sine

Related Wolfram sites

https://functions.wolfram.com/ElementaryFunctions/ArcSin/

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References

Abramowitz, M. and Stegun, I. A.(Eds.). "Inverse Circular Functions." §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.Apostol, T. M. Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 253-254, 1967.Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, 2007.Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143 and 220, 1987.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Borwein, J. M. and Chamberland, M. "Integer Powers of Arcsin." Int. J. Math. Math. Sci., Art. 19381, 1-10, 2007.Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 3rd ed. New York: Springer-Verlag, pp. 69-70, 1997.GNU C Library. "Mathematics: Inverse Trigonometric Functions." https://sourceware.org/glibc/manual/latest/html_node/Inverse-Trig-Functions.html.Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 307, 1998.Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions." §2.7 in Handbook of Mathematical Formulas and Integrals, 2nd ed. Orlando, FL: Academic Press, pp. 124-128, 2000.Sloane, N. J. A. Sequences A002595/M4233 and A055786 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.Zwillinger, D.(Ed.). "Inverse Circular Functions." §6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 465-467, 1995.

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Inverse Sine

Cite this as:

Weisstein, Eric W. "Inverse Sine." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseSine.html

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