 Hi and welcome to the session. Today we will learn about principal value branches of inverse trigonometric functions. As we already know that for a function f, f inverse exists if f is 1, 1 on 2. Also we know that trigonometric functions are not 1, 1 on 2 over their natural domains ranges. So this implies their inverse does not exist. If we restrict their domains ranges then they will become 1, 1. This implies their inverse will exist over those domains and ranges. So let us study about the domains and ranges of inverse trigonometric functions. As we already know that the domain of sine inverse function is the closed interval from minus 1 to 1 and the range of principal value branch is the closed interval from minus pi by 2 to pi by 2. So here our function is sine inverse its domain is the closed interval from minus 1 to 1 and the range of principal value branch is the closed interval from minus pi by 2 to pi by 2. Now we will consider cos inverse function. We already know that for cos function its domain is the set of all real numbers and its range is the closed interval from minus 1 to 1. If we restrict the domain cos function to any of these intervals that is closed interval from minus pi to 0 or closed interval from 0 to pi or closed interval from pi to 2 pi etc. then cos function will become 1, 1 on 2 range as a closed interval from minus 1 to 1. So this implies the inverse of cos function denoted as cos inverse exists with domain as a closed interval from minus 1 to 1 and range as any of these class intervals. Now corresponding to each such interval we get a branch of the function cos inverse but the branch with the range as a closed interval from 0 to pi is called the principal value branch of the function cos inverse. So here the function is cos inverse its domain is the closed interval from minus 1 to 1 and the range of principal value branch is a closed interval from 0 to pi. Now let's move on to cos inverse function. As we know that cos x is the inverse of sin x that is 1 upon sin x and the domain of cos function is the set of numbers say x such that x belongs to set of real numbers and x is not equal to n pi where n belongs to z and its range is the set of real numbers except the open interval from minus 1 to 1. If we restrict the domain of cos function to any of these intervals then cos function becomes 1 1 and on to with range as the set of real numbers except the open interval from minus 1 to 1. So this means that the inverse of cos function denoted as cos inverse exists and corresponding to each such interval we get a branch of the function cos inverse and the branch with range as a closed interval from minus pi by 2 to pi by 2 except 0 is called the principal value branch of the function cos inverse. So the domain of the function cos inverse is the set of real numbers except the open interval from minus 1 to 1 and the range of principal value branch is a closed interval from minus pi by 2 to pi by 2 except 0. Next we will learn about secant inverse function. Secant x is the reciprocal of cos x that is 1 upon cos x. Now the domain of secant function is the set of real numbers except the set of numbers say x such that x is equal to 2n plus 1 into pi by 2 where n belongs to z and its range is the set of real numbers except the open interval from minus 1 to 1. So if we restrict the domain of secant function to any of these intervals then the secant function becomes 1 1 on to with range as the set of real numbers except the open interval from minus 1 to 1 this implies that secant inverse exists with domain as the set of real numbers except the open interval from minus 1 to 1 and range could be any of these intervals. Also corresponding to each of these intervals we get a different branch for the function secant inverse but the branch with the range as the closed interval from 0 to pi except pi by 2 is the principal value branch of the function secant inverse. So for the function secant inverse the domain is the set of real numbers except the open interval from minus 1 to 1 and the range of principal value branch is the closed interval from 0 to pi except pi by 2. Now the next function is secant inverse function. The domain of secant function is the set of numbers say x such that x belongs to r and x is not equal to 2n plus 1 into pi by 2 where n belongs to z and its range is the set of real numbers. If we restrict the domain of tan function to any of these intervals then the tan function becomes 1 1 on to with range as the set of real numbers. So this implies inverse of tan function denoted as tan inverse exists with domain as the set of real numbers and range could be any of these intervals. Also corresponding to each of these intervals we get a different branch of the function tan inverse but the branch corresponding to the interval minus pi by 2 to pi by 2 is called the principal value branch of the function tan inverse. So here for the function tan inverse the domain is the set of real numbers and range of the principal value branch is the open interval from minus pi by 2 to pi by 2. Now our last function is cot inverse function and cot x is equal to 1 upon tan x. The domain of cot function is the set of numbers say x such that x belongs to r and x is not equal to n pi where n belongs to z and its range is the set of real numbers. If we restrict the domain of cot function to any of these intervals then the cot function becomes 1 1 on to with range as the set of real numbers. So this implies the function cot inverse exists with domain as the set of real numbers and range could be any of these intervals. Also corresponding to each of these intervals we get a different branch of the function cot inverse but the branch with range as open interval from 0 to pi is called the principal value branch. So for the function cot inverse the domain is the set of real numbers and the range of principal value branch is the open interval from 0 to pi. So with this we finish this session hope you must have understood all the concepts. Goodbye take care and have a nice day.