 Hello and welcome to this session. In this session, we will discuss the graphs of the inverse trigonometric functions, trigonometric functions and cot-inverse x. First of all, let us discuss the graph of the inverse x. Now, the trigonometric functions are in general not 1, 1, and 2. Therefore, their inverses do not exist, but if we restrict their domains, then they become 1, 1, and 2, and hence we can have their inverses. Now, let us define a function f from set of real numbers r to the open interval minus pi by 2 to pi by 2 by f of x is equal to restricted for domain to open interval minus pi by 2 to pi by 2. Therefore, the domain is a set of real numbers, that is, the possible values of x for the graph of the inverse x and the inverse x is defined in the open interval minus infinity to infinity, so we can write that the domain of the inverse x is the open interval minus infinity to infinity and the range of term inverse x is the written interval minus pi by 2 to pi by 2. Now, let y is equal to f of x is equal to term inverse x, that is, let y is equal to term inverse x. Now, let us draw the graph of y is equal to term inverse x. Now, y is equal to term inverse x implies x is equal to term y. Now, for the different values of x and y, first of all, let us draw a table. Now, we have x is equal to term y, so for y is equal to minus pi by 3, x is equal to term minus pi by 3, which is equal to minus rule 3, which is equal to minus 1.73. Therefore, for y is equal to minus pi by 3, minus 1.73. And now, for y is equal to minus pi by 4, which is equal to minus 1. So, for y is equal to minus pi by 4, x is equal to minus 1 to minus pi by 6, x is equal to term of minus pi by 6, which is equal to minus 0.58. So, for y is equal to minus pi by 4 to minus 0.5 is equal to 0. We are getting x is equal to term 0, which is 0. So, for y is equal to 0, x is equal to 0. Then, for y is equal to pi by 6, x is equal to 1 by rule 3, which is equal to 0.58. So, for y is equal to pi by 6, x is equal to 0.58. And then, for y is equal to pi by 4, which is 1. So, for y is equal to pi by 4, x is equal to 1. And now, for y is equal to pi by 3, which is equal to rule 3, which is equal to 1.73. So, for y is equal to pi by 3, x is equal to 1.73. Now, for drawing the graph of y is equal to 10 in the various values of x. And here, y is equal to 10 inverse m. Now, in the graph, we have taken 10 small divisions is equal to 1 along the x-axis. Large divisions is equal to pi by 6 across point on the graph, that is minus 1.70 minus pi by 3 on the graph. So, this is the required point on the graph. Now, we will plot the next point, that is minus 1. Now, this is the required point on the graph. We will plot this on the graph. So, we have plotted all the points on the graph. Now, on joining all these points, we are getting the graph of y is equal to 10 inverse m. Now, here you can see that as y increases monotrally from minus pi by 2 to pi by 2, it increases from minus x is 0, is 0 defined in the open interval minus infinity to plus infinity. So, 10 inverse x tends to pi by 2, which is the curve of y is equal to 10 inverse x plus infinity, that is y, turns to minus pi by 2 is infinity. Now, let us discuss the graph of cot inverse x 0 to pi is equal to cot inverse x. We have restricted the domain of cot x to the open interval 0 to pi. Therefore, the domain is a factor of real numbers, that is, the category x values for the graph of cot inverse x have an interval minus infinity to infinity. So, we can write the domain of cot inverse x has an interval minus infinity to infinity 0 to pi where the graph of cot inverse x tends to infinity to minus infinity. Then cot inverse 0 is equal to y is equal to f of x is equal to cot inverse x, that is, let y is equal to cot inverse x. Let us draw the graph of y is equal to cot inverse m. Now, we have y is equal to cot inverse x cot y. Now, for this equation, we will find the ways values of x and y. For this, let us draw a table. Now, we have drawn a table. Now, we know that trigonometric function cot inverse x is an open interval 0 to pi. So, we will choose the ways values of y in between 0 and a pi. So, for y is equal to minus root 3 which is equal to minus 1.73. Now, for y is equal to pi over 4 which is equal to minus 1. Now, for y is equal to minus 1 by root 3 which is equal to minus 0.5 i is equal to pi by 2 equal to cot pi by 2 which is 0. It is 1 by root 3 which is equal to 0.5 which is 1 to 1.73. Now, i is equal to cot inverse x. We will plot these points. Now, in the graph we have taken 10 small divisions is equal to 1 along the x-axis and 5 small divisions is equal to pi by 6 along the y-axis. We have got these points on the graph. First of all, we will plot the point minus 1.75 point on the graph point minus 1, 3 pi by 4 on the graph. Now, this is the required point on the graph. We will plot all the points on the graph. So, we have plotted all the points on the graph. Now, on drawing all these points we are getting the graph of y is equal to cot inverse n currently from 0 to 0 is equal to pi by 3 that is the x-axis whose equation is y is equal to 0 and this is the line whose equation is y is equal to minus y. f of y is equal to cot inverse n and in this session we have learnt about the graphs of y is equal to 10 inverse x and y is equal to cot inverse x. So, this is the principle.