Inverse Trigonometric Function

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inverse trigonometric function

[¦in‚vərs ‚trig·ə·nə‚me·trik ′fəŋk·shən]
(mathematics)
An inverse function of a trigonometric function; that is, an arc sine, arc cosine, arc tangent, arc cotangent, arc secant, or arc cosecant. Also known as antitrigonometric function.

Inverse Trigonometric Function

a function that is a solution of the problem of finding an arc (number) from a given value of its trigonometric function. The six inverse trigonometric functions correspond to the six trigonometric functions: (1) Arc sin x, the inverse sine of x; (2) Arc cos x, the inverse cosine of x; (3) Arc tan x, the inverse tangent of x; (4) Arc cot x, the inverse cotangent of x; (5) Arc sec x, the inverse secant of x; and (6) Arc csc x, the inverse cosecant of x.

As an example, according to these definitions, x = Arc sin a is any solution of the equation sin x = a; that is, sin Arc sin a = a. The functions Arc sin x and Arc cos x are defined in the real domain for ǀxǀ ≤ 1, the functions Arc tan x and Arc cot x are defined for all real x, and the functions Arc sec x and Arc csc x are defined for ǀxǀ ≥ 1. The last two functions are rarely used.

Since trigonometric functions are periodic, their inverse functions are multiple-valued. Certain single-valued branches (the principal branches) of these functions are designated as arc sin x, arc cos x, …, arc csc x. Specifically, arc sin x: is the branch of the function Arc sin x for which –π/2 ≤ arc sin x ≤ π/2. Similarly, the functions arc cos x, arc tan x, and arc cot x are determined, respectively, from the conditions 0 < arc cos x ≤ π, –π/2 < arc tan x < π/2, and 0 < arc cot x < π. The inverse trigonometric functions Arc sin x, … can easily be expressed in terms of arc sin x, …; for example,

Arc sin x = (—1)n arc sin x + πn

Arc cos x = ±arc cos x + 2πn

Arc tan x = arc tan x + πn

Arc cot x = arc cot x + πn

where n = 0, ±1, ±2, ….

The well-known relations between the trigonometric functions yield relations between the inverse trigonometric functions, for example, the formula

implies that

The derivatives of the inverse trigonometric functions have the form

The inverse trigonometric functions can be represented as power series, for example,

These series converge for – 1 ≤ x ≤ 1.

The inverse trigonometric functions can be determined for arbitrary complex values of the independent variable, but the values of these functions will be real only for the values of the independent variable indicated above. Inverse trigonometric functions of a complex independent variable can be expressed by means of a logarithmic function, for example,

REFERENCE

Novoselov, S. I. Obratnye trigonometricheskie funktsii, 3rd ed. Moscow, 1950.
References in periodicals archive ?
Two-dimensional curve is interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions.
Proposed method of Hurwitz-Radon Matrices (MHR) is applied in curve modeling via different coefficients: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arcsin, arccos, arctan, arcctg or power.
Coefficients for curve modeling are computed using probability distribution functions: polynomials, power functions, sinus, cosine, tangent, cotangent, logarithm, exponent or arcsin, arccos, arctan, arcctg.
Coefficient [gamma] is calculated using different functions (polynomials, power functions, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan or arcctg, also inverse functions) and choice of function is connected with initial requirements and curve specifications.
[gamma] = [(2 - 4/[pi] x arcctg [alpha]).sup.s], s > 0.
which is chosen by the condition 0 [much less than] arcctg ([I.sub.e]/[I.sub.t]) [much less than] [pi].
[I.sub.m] = [square root of ([I.sup.2.sub.t] + [I.sup.2.sub.e])] [J.sub.m] (arcctg ([I.sub.e]/[I.sub.t])) [i.sup.m] exp(im [psi]), (46)
[I.sub.m=0] = [square root of ([I.sup.2.sub.t] + [I.sup.2.sub.e])] [J.sub.0] (arcctg ([I.sub.e]/[I.sub.t]).
In the case, when arcctg [[I.sub.e]/[I.sub.t]] = 2.4 (at z = 2.4, the Bessel function equals zero [J.sub.0](z) = 0, that implies [I.sub.m=0] = 0.
[I.sub.m=1] = i[square root of ([I.sup.2.sub.t] + [I.sup.2.sub.e)] [J.sub.1] (arcctg (exp([I.sub.e]/[I.sub.t])), (48)

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