we take it as the function whose domain is [– , ] and range is . We write sin – : [– , ] → From the definition of the inverse functions, it follows that sin (sin – x ) = x if – ≤ x ≤ and sin – (sin x ) = x if . In other words, if y = sin – x , then sin y = x . Remarks (i) We know from Chapter , that if y = f ( x ) is an invertible function, then x = f – ( y ).
Thus, the graph of sin – function can be obtained from the graph of original function by interchanging x and y axes, i.e., if ( a , b ) is a point on the graph of sine function, then ( b , a ) becomes the corresponding point on the graph of inverse of sine function. Thus, the graph of the function y = sin – x can be obtained from the graph of y = sin x by interchanging x and y axes. The graphs of y = sin x and y = sin – x are as given in Fig . (i), (ii), (iii).
The dark portion of the graph of y = sin – x represent the principal value branch. (ii) It can be shown that the graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image (i.e., reflection) along the line y = x . This can be visualised by looking the graphs of y = sin x and y = sin – x as given in the same axes (Fig . (iii)).
Like sine function, the cosine function is a function whose domain is the set of all real numbers and range is the set [– , ]. If we restrict the domain of cosine function to [ , π ], then it becomes one-one and onto with range [– , ]. Actually, cosine function Fig . (ii) Fig .
(iii) Fig . (i) INVERSE TRIGONOMETRIC FUNCTIONS