What is Inverse Trigonometric function ?

Inverse Trigonometric function

In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.

There are several notations used for the inverse trigonometric functions.

The most common convention is to name inverse trigonometric functions using an arc- prefix: arcsin(x), arccos(x), arctan(x), etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: When measuring in radians, an angle of θradians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus, in the unit circle, "the arc whose cosine is x" is the same as "the angle whose cosine is x", because the length of the arc of the circle in radii is the same as the measurement of the angle in radians. In computer programming languages the inverse trigonometric functions are usually called by the abbreviated forms asin, acos, atan.

The notations sin⁻¹(x), cos⁻¹(x), tan⁻¹(x), etc., as introduced by John Herschel in 1813, are often used as well in English-language sources and this convention complies with the notation of an inverse function. This might appear to conflict logically with the common semantics for expressions like sin²(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of the reciprocal trigonometric functions has its own name—for example, (cos(x))⁻¹ = sec(x). Nevertheless, certain authors advise against using it for its ambiguity. Another convention used by a few authors is to use a majuscule (capital/upper-case) first letter along with a −1 superscript: Sin⁻¹(x), Cos⁻¹(x), Tan⁻¹(x), etc. This potentially avoids confusion with the multiplicative inverse, which should be represented by sin⁻¹(x), cos⁻¹(x), etc.