Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

Additional Topics for IIT JAM Mathematics

Mathematics : Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

The document Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev is a part of the Mathematics Course Additional Topics for IIT JAM Mathematics.
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The inverse trigonometric functions are the inverse functions of the trigonometric functions, written cos-1z, cot-1z, csc-1z, sec-1z, sin-1z, and tan-1z.
Alternate notations are sometimes used, as summarized in the following table.

f(z)

alternate notations

cos-1 z

arccos z (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)

cot-1 z

arccot z (Spanier and Oldham 1987, p. 333), arcctg z (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 208; Jeffrey 2000, p. 127)

csc-1 z

arccsc z (Spanier and Oldham 1987, p. 333), arccosec z (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)

sec-1 z

arcsec z (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 209)

sin-1 z

arcsin z (Spanier and Oldham 1987, p. 333; Gradshteyn and Ryzhik 2000, p. 207)

tan-1 z

arctan z (Spanier and Oldham 1987, p. 333), arctg z (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 Arcsin z (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

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcCsc[z]

(-1, 1)

inverse cosine

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcCos[z]

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev 

inverse cotangent

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcCot[z]

(-i, i)

inverse secant

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcSec[z]

(-1, 1)

inverse sine

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcSin[z]

 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev 

inverse tangent

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

ArcTan[z]

 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev 

Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
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 lineBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev illustrated above.

function namefunctiondomainrange
inverse cosecantBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev(-∞, ∞) Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
inverse cosineBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev[-1, 1]Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
inverse cotangentBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev(-∞, ∞)Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
inverse secantBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev(-∞, ∞)Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
inverse sineBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev[-1, 1]Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
inverse tangentBaics of Inverse Trigonometric Functions Mathematics Notes | EduRev(-∞, ∞)Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

Inverse-forward identities are

 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

Forward-inverse identities are
 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
Inverse sum identities include
 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev
where equation (11) is valid only for x >0.
Complex inverse identities in terms of natural logarithms include
 Baics of Inverse Trigonometric Functions Mathematics Notes | EduRev

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