 Hello and welcome to the session. In this session we will discuss the graphs of the inverse trigonometric functions, secant inverse x and cosecant inverse x. First of all let us discuss the graph of secant inverse x. Now trigonometric functions are in general not 1, 1, 2. Therefore the inverse do not exist but if we restrict the domains then they become 1, 1, 2 and hence we can have the inverses. Now let us define a function f from the set of real numbers excluding the upper interval minus 1 to 1 to the close interval 0 to pi excluding pi by 2 by is equal to secant inverse x. Now the domain of secant inverse x is a set of real numbers excluding the upper interval minus 1 to 1. That is x is defined for x is lesser equal to minus 1 is greater than equal to 1. Its range is the close interval 0 to pi excluding pi by 2. Now let y is equal to f of x is equal to secant inverse x that is let y is equal to secant inverse x. Now let us draw the graph of y is equal to secant inverse x. Now y is equal to secant inverse x implies x is equal to secant inverse y. Now first of all for the different values of x and y let us draw a table. So we have drawn a table. Now in this equation y is equal to 0, x is equal to secant 0 which is 1 equal to 0, x is equal to 1. Then for y is equal to pi by 6 x is equal to secant inverse x which is equal to 2 by root 3 which is equal to 1 point. So for y is equal to pi by 6 x is equal to 1.15. Now for y is equal to pi by 4 x is equal to secant pi by 4 which is equal to root 2 which is equal to 1.41. So for y is equal to pi by 4 is equal to 1.41. Now for y is equal to pi by 3 which is equal to so for y is equal to pi by 3 x is equal to 2, y is equal to 2 pi by 3 which is minus equal to 2 pi by 3 is equal to minus 2. Now for y is equal to pi by 4 is equal to minus root 2 1.41. So for y is equal to minus 1.41. And now to secant pi by pi by 6 is equal to minus 2 by root 3 which is equal to minus 1.15. So for y is equal to pi by 6 is equal to minus 1.15 which is minus 1. Now for drawing the graph of y is equal to secant inverse x we will plot on the graph. 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. For drawing the graph first of all we will plot the point 1 0 on the graph. Now this is the required point on the graph. Now we will plot the point 1.15 and pi by 6 on the graph. Now this is the required point on the graph all the points on the graph. So we have plotted all the points on the graph. Now of this line is y is equal to pi and the equation of the x axis is y is equal to 0. This equation is y is equal to pi by 2. Now by drawing the points which we have plotted we are getting the graph that y increases from pi by 2 minus 1. Inverse minus 1 is pi less than equal to minus 1. We can say that secant inverse x is the angle line between 0. Now let us define minus 1 to 1 to the closed interval minus pi by 2 to pi by 2 excluding 0. By f of x is equal to secant inverse f of the interval minus 1 to 1. f of less than equal to minus 1 equal to 1 to pi by 2 excluding 0 to 0. From the positive values turns to infinity to 0. From the negative values is equal to let y is equal to secant inverse x. Let us draw the graph of y is equal to secant inverse x. Now y is equal to secant inverse x implies x is equal to secant y. Now let us draw a table for the different values of x and y values of x from this function equation. Suppose y is equal to 1.41. Now for y is equal to 2 over root 3 which is 1.15. y is equal to pi by 2, x is equal to secant pi by 2 which is 1. Then for y is equal to minus inch minus pi by 6 which is minus 2 which is minus 1.14. y is equal to minus plus is equal to minus 1.15 minus 1. Now for drawing the graph of y is equal to secant inverse x. We will plot all these points on the graph. Now again for the graph we have taken is equal to 1 along the x axis and 5 small divisions is equal to pi by 6. Let us plot the point 2 pi by 6 on the graph. We have plotted this point on the graph for the point 1.41 with other point on the graph. We will plot all the points on the graph. We have plotted all the points on the graph. Now by drawing all these points we are getting the graph of y is equal to secant inverse x. And here this is the line whose equation is y is equal to pi by 2. And the equation of x axis is y is equal to 0. And this is the line whose equation is y is equal to 0 to minus pi increases from y decreases to 0 increases. And cosecant inverse x from the graph we can also say that cosecant inverse x is the angle. Line letters cosecant inverse x which is equal to y whose cosecant you have learnt about. The graph of y is equal to cosecant inverse x and y is equal to cosecant inverse x. And this completes our session. Hope you all have enjoyed the session.