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Inverse Trigonometric Functions

The inverse trigonometric functions are the inverse functions of the trigonometric functions, written , , , , , and .

Alternate notations are sometimes used, as summarized in the following table.

 alternate notations (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 209) (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207) (Spanier and Oldham 1987, p. 333), (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127)

The inverse trigonometric functions are multivalued. For example, there are multiple values of such that , so is not uniquely defined unless a principal value is defined. Such principal values are sometimes denoted with a capital letter so, for example, the principal value of the inverse sine may be variously denoted or (Beyer 1987, p. 141). On the other hand, the notation (etc.) is also commonly used denote either the principal value or any quantity whose sine is an (Zwillinger 1995, p. 466). Worse still, the principal value and multiply valued notations are sometimes reversed, with for example denoting the principal value and denoting the multivalued functions (Spanier and Oldham 1987, p. 333).

Since the inverse trigonometric functions are multivalued, they require branch cuts in the complex plane. Differing branch cut conventions are possible, but those adopted in this work follow those used by the Wolfram Language, summarized below.

 function name function Wolfram Language branch cut(s) inverse cosecant ArcCsc[z] inverse cosine ArcCos[z] and inverse cotangent ArcCot[z] inverse secant ArcSec[z] inverse sine ArcSin[z] and inverse tangent ArcTan[z] and

Different conventions are possible for the range of these functions for real arguments. Following the convention used by the Wolfram Language, the inverse trigonometric functions defined in this work have the following ranges for domains on the real line , illustrated above.

 function name function domain range inverse cosecant or inverse cosine inverse cotangent or inverse secant or inverse sine inverse tangent

Inverse-forward identities are

 (1) (2) (3)

Forward-inverse identities are

 (4) (5) (6) (7) (8) (9)

Inverse sum identities include

 (10) (11) (12)

where equation (11) is valid only for .

Complex inverse identities in terms of natural logarithms include

 (13) (14) (15)

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

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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. "Inverses of the Trigonometric Functions." §6.21 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 253-256, 1967.Beyer, W. H.(Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, 1987.Harris, J. W. and Stocker, H. "Inverse Trigonometric Functions." Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 306-318, 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.Spanier, J. and Oldham, K. B. "Inverse Trigonometric Functions." Ch. 35 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 331-341, 1987.Trott, M. "Inverse Trigonometric and Hyperbolic Functions." §2.2.5 in The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 180-191, 2004. http://www.mathematicaguidebooks.org/.Zwillinger, D.(Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.

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Inverse Trigonometric Functions

Cite this as:

Weisstein, Eric W. "Inverse Trigonometric Functions." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseTrigonometricFunctions.html