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)

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.