restricted to any of the intervals [– π , ], [ , π ], [ π , π ] etc., is bijective with range as [– , ]. We can, therefore, define the inverse of cosine function in each of these intervals. We denote the inverse of the cosine function by cos – (arc cosine function). Thus, cos – is a function whose domain is [– , ] and range could be any of the intervals [– π , ], [ , π ], [ π , π ] etc.
Corresponding to each such interval, we get a branch of the function cos – . The branch with range [ , π ] is called the principal value branch of the function cos – . We write cos – : [– , ] → [ , π ]. The graph of the function given by y = cos – x can be drawn in the same way as discussed about the graph of y = sin – x .
The graphs of y = cos x and y = cos – x are given in Fig . (i) and (ii). Fig . (ii) Let us now discuss cosec – x and sec – x as follows: Since, cosec x = sin x , the domain of the cosec function is the set { x : x ∈ R and x ≠ n π , n ∈ Z } and the range is the set { y : y ∈ R , y ≥ or y ≤ – } i.e., the set R – (– , ).
It means that y = cosec x assumes all real values except – < y < and is not defined for integral multiple of π . If we restrict the domain of cosec function to – { }, then it is one to one and onto with its range as the set R – (– , ). Actually, cosec function restricted to any of the intervals , { } −π −π −−π – { }, { } −π etc., is bijective and its range is the set of