 Hi and welcome to the session. Let us learn what do we mean by principle value branch. We know that the inverse of a function 8 exists if the function is 1, 1, and onto. But, trigonometric functions do not 1, 1, and onto. Their inverse exists if we restrict the domain and ranges of the trigonometric functions. Let us first discuss the sine function. We know that the domain of sine function is the set of all real numbers and its range is closed interval minus 1, 1. If we restrict the domain r to close interval minus pi by 2 to pi by 2, we find that the sine function becomes 1, 1, onto and then its range would be close interval minus 1 to 1. In fact, if we restrict the domain to any of the following intervals, we find that sine function becomes 1, 1, onto. We can say sine inverse exists and the domain of sine inverse function would be close interval minus 1 to 1 and its range would be any of these closed intervals that is minus 3 pi by 2 to minus pi by 2 or minus pi by 2 to pi by 2 or pi by 2 to 3 pi by 2. Now, let us learn whatever principal value branch. Corresponding to each interval, we get different branches of sine inverse. The branch with range closed interval minus pi by 2 to pi by 2 is called the principal value branch, the function sine inverse. Other intervals as range give different branches of sine inverse. Now, let us learn the principal value branch of sine inverse function with the help of graph. This is the graph of y is equal to sine x. We can obtain the graph of y is equal to sine inverse x with the help of y is equal to sine x by interchanging the x and y axis. The curve between minus 1 and 1 as we can see is called the principal value branch for sine inverse function. So, this is the required principal value branch of sine inverse function. Similarly, we find the principal value branch of other inverse trigonometric function. So, hope you have understood this module. Have a nice day. Bye-bye.