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)