Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under 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.

Notation

There are several notations for the inverse trigonometric functions. The most common uses the prefix arc-: 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 θ radians will correspond to an arc whose length is rθ, where r is the radius of the circle. Thus in the unit circle, the cosine of x function is both the arc and the angle, because the arc of a circle of radius 1 is the same as the angle. Or, "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 often called by the abbreviated forms asin, acos, atan.

Another common notation in English sources uses a superscript f−1: sin−1(x), cos−1(x), tan−1(x), etc., as introduced by John Herschel in 1813.