FUNCTIONS Aryabhata ( - A. D.) INVERSE TRIGONOMETRIC FUNCTIONS We have also learnt in Chapter that if f : X → Y such that f ( x ) = y is one-one and onto, then we can define a unique function g : Y → X such that g ( y ) = x , where x ∈ X and y = f ( x ), y ∈ Y. Here, the domain of g = range of f and the range of g = domain of f . The function g is called the inverse of f and is denoted by f – .
Further, g is also one-one and onto and inverse of g is f . Thus, g – = ( f – ) – = f . We also have ( f – o f ) ( x ) = f – ( f ( x )) = f – ( y ) = x and ( f o f – ) ( y ) = f ( f – ( y )) = f ( x ) = y Since the domain of sine function is the set of all real numbers and range is the closed interval [– , ]. If we restrict its domain to , then it becomes one-one and onto with range [– , ].
Actually, sine function restricted to any of the intervals , etc., is one-one and its range is [– , ]. We can, therefore, define the inverse of sine function in each of these intervals. We denote the inverse of sine function by sin – (arc sine function). Thus, sin – is a function whose domain is [– , ] and range could be any of the intervals , −π −π or , and so on.
Corresponding to each such interval, we get a branch of the function sin – . The branch with range is called the principal value branch, whereas other intervals as range give different branches of sin – . When we refer to the function sin – ,