Notes to a video lecture on http://www.unizor.com

Inverse Trigonometric Functions - ARCCOT

The original cotangent function defined for any real argument does not have an inverse function because it does not establish a one-to-one correspondence between its domain and a range.
To be able to define an inverse function, we have to reduce the original definition of a cotangent function to an interval where this correspondence does take place. Any interval where cotangent is monotonic and takes all values in its range would fit this purpose.

For a function y=cot(x) an interval of monotonic behavior is usually chosen as (0,π), where the function is monotonously decreasing from +∞ to −∞.
This variant of a cotangent function, reduced to an interval where it is monotonous and fills an entire range, has an inverse function.

The inverse function to it is called arccotangent and is symbolized as y=arccot(x).

Its domain is an interval (−∞,+∞) (same as a range of y=cot(x)) where it is monotonously decreasing from π to 0 (same as a domain of y=cot(x)).

Its codomain and a range is an interval (0,π) (same as a domain of y=cot(x)).

Graphs of functions y=cot(x) with a domain (0,π) and a range (−∞,+∞) and y=arccot(x) with a domain (−∞,∞) and a range (0,π) are symmetrical relative to a bisector of a main coordinate angle between positive directions of X-axis and Y-axis.