We know that domain of the cot function (cotangent function) is the set {x : x ∈ R and x ≠ nπ, n ∈ Z} and range is R. It means that cotangent function is not defined for integral multiples of π. If we restrict the domain of cotangent function to (0, π), then it is bijective and its range as R. In fact, cotangent function restricted to any of the intervals (-π, 0), (0, π), (π, 2π) etc., is bijective and its range is R. Thus cot-1 can be defined as a function whose domain is the R and range as any of the intervals (- π, 0), (0, π), (π, 2π) etc. These intervals give different branches of the function cot-1. The fimction with range (0, π) is called the principal value branch of the function cot-1. We thus have
cot-1 : R → (0, π)
The graphs of y = cot x and y = cot-1 x are given in (a) and (b)
To Remember :
The following table gives the inverse trigonometric function (principal value branches) along with their domains and ranges.
Note :
- sin-1 x should not be confused with (sin x)-1. In fact (sin x)-1 ≠ \(\frac{1}{sin\ x}\) and similarly for other trigonometric functions.
- Whenever no branch of an inverse trigonometric functions is mentioned, we mean the principal value branch of that function.
- The value of an inverse trigonometric functions lies in the range of principal value of that inverse trigonometric functions.
