arccotx

arccotx, also known as arccotangent or inverse cotangent, is a mathematical function that represents the inverse of the cotangent function. For a given real number x, arccotx is the angle whose cotangent is x. The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side in a right-angled triangle, or as the ratio of cosine to sine of an angle.

The domain of arccotx is all real numbers, meaning it can accept any real number as input. However, to ensure that arccotx is a well-defined function (i.e., each input has a unique output), its range is typically restricted. The principal value of arccotx is usually taken to be in the open interval (0, pi) radians, or (0, 180) degrees. This means that for any real number x, arccotx will return an angle between 0 and pi radians, exclusive.

The derivative of arccotx is given by -1 / (1 + x^2). This derivative is important in calculus for integrating functions involving arccotx or its related forms. The integral of arccotx is x arccotx - (1/2) ln(1 + x^2) + C, where C is the constant of integration.

Arccotx can be expressed in terms of arctanx using the relationship arccotx = pi/2 - arctanx for all real x. This identity is useful because the arctangent function is more commonly encountered and its properties are often more familiar. Graphs of y = arccotx show a continuous, decreasing function that approaches the horizontal asymptotes y = 0 as x approaches positive infinity and y = pi as x approaches negative infinity.