Inverse Trigonometric Function(redirected from Arc functions)
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inverse trigonometric function[¦in‚vərs ‚trig·ə·nə‚me·trik ′fəŋk·shən]
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
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,