INVERSE TRIGONOMETRIC FUNCTIONS
he inverse trigonometric functions are the inverse functions of the trigonometric functions, written cos^(-1)z, cot^(-1)z, csc^(-1)z, sec^(-1)z, sin^(-1)z, and tan^(-1)z.
Alternate notations are sometimes used, as summarized in the following table.
f(z) alternate notations
cos^(-1)z arccosz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)
cot^(-1)z arccotz (Spanier and Oldham 1987, p. 333), arcctgz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127)
csc^(-1)z arccscz (Spanier and Oldham 1987, p. 333), arccosecz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)
sec^(-1)z arcsecz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 209)
sin^(-1)z arcsinz (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)
tan^(-1)z arctanz (Spanier and Oldham 1987, p. 333), arctgz (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 w such that z=sinw, so sin^(-1)z 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 sin^(-1)z may be variously denoted Sin^(-1)z or Arcsinz (Beyer 1987, p. 141). On the other hand, the notation sin^(-1)z (etc.) is also commonly used denote either the principal value or any quantity whose sine is z an (Zwillinger 1995, p. 466). Worse still, the principal value and multiply valued notations are sometimes reversed, with for example arcsinz denoting the principal value and Arcsinz 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 csc^(-1)z ArcCsc[z] (-1,1)
inverse cosine cos^(-1)z ArcCos[z] (-infty,-1) and (1,infty)
inverse cotangent cot^(-1)z ArcCot[z] (-i,i)
inverse secant sec^(-1)z ArcSec[z] (-1,1)
inverse sine sin^(-1)z ArcSin[z] (-infty,-1) and (1,infty)
inverse tangent tan^(-1)z ArcTan[z] (-iinfty,-i] and [i,iinfty)
InverseTrigonometricFunctions
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 R, illustrated above.
function name function domain range
inverse cosecant csc^(-1)x (-infty,infty) [-1/2pi,0) or (0,1/2pi]
inverse cosine cos^(-1)x [-1,1] [0,pi]
inverse cotangent cot^(-1)x (-infty,infty) (-1/2pi,0) or (0,1/2pi]
inverse secant sec^(-1)x (-infty,infty) [0,1/2pi) or (1/2pi,pi]
inverse sine sin^(-1)x [-1,1] [-1/2pi,1/2pi]
inverse tangent tan^(-1)x (-infty,infty) (-1/2pi,1/2pi)
Inverse-forward identities are
tan^(-1)(cotx) = 1/2pi-x forx in [0,pi]
sin^(-1)(cosx) = 1/2pi-x forx in [0,pi]
sec^(-1)(cscx) = 1/2pi-x forx in [0,1/2pi].
Forward-inverse identities are
cos(sin^(-1)x) = sqrt(1-x^2)
cos(tan^(-1)x) = 1/(sqrt(1+x^2))
sin(cos^(-1)x) = sqrt(1-x^2)
sin(tan^(-1)x) = x/(sqrt(1+x^2))
tan(cos^(-1)x) = (sqrt(1-x^2))/x
tan(sin^(-1)x) = x/(sqrt(1-x^2)).
Inverse sum identities include
sin^(-1)x+cos^(-1)x = 1/2pi
tan^(-1)x+cot^(-1)x = 1/2pi
sec^(-1)x+csc^(-1)x = 1/2pi,
where equation (11) is valid only for x>=0.
Complex inverse identities in terms of natural logarithms include
sin^(-1)z = -iln(iz+sqrt(1-z^2))
cos^(-1)z = 1/2pi+iln(iz+sqrt(1-z^2))
tan^(-1)z = 1/2i[ln(1-iz)-ln(1+iz)].
