 Hello and welcome to the session. In this session we will discuss the graphs of the inverse trigonometric functions, sine inverse x and cos inverse x. Now first of all let us discuss the graph of sine inverse x. Now we know that trigonometric functions are in general not 1 1 on 2, therefore their inverses do not exist. But if we restrict their domains then they become 1 1 on 2 and hence we can have their inverses. Now let us define a function from the closed interval minus 1 to run 2 inverse interval minus pi by 2 to pi by 2 by f of x is equal to sine inverse x. That is we have restricted the domain of sine x to the closed interval minus pi by 2 to pi by 2. Therefore the domain of sine inverse x is the closed interval minus 1 to 1 that is equal to minus 1 and less than equal to 1. We can also say that sine inverse x is defined in the closed interval minus 1 to 1 only and the range of sine inverse x is the closed interval minus pi by 2 to pi by 2. That is the possible values of f of x for the graph of sine inverse x will be greater than equal to minus pi by 2 and less than equal to pi by 2. Now let y is equal to f of x is equal to sine inverse x that is y is equal to sine inverse x. Now let us draw the graph of y is equal to sine inverse x. Now y is equal to sine inverse x implies x is equal to sine y. Now for different values of x and y let us draw a table. Now for y is equal to minus pi by 2 x will be equal to sine of minus pi by 2 which is equal to minus 1. Then for y is equal to minus pi by 3 x is equal to sine of minus pi by 3 which is minus root 3 by 2 is equal to minus 0.87. For y is equal to minus pi by 4 x is equal to sine of minus pi by 4 which is equal to minus 1 by root 2 which is minus 0.71. Then for y is equal to minus pi by 6 pi by 6 which is minus 1 by 2 which is equal to minus 0.5. For y is equal to 0 x is equal to 0 then for y is equal to pi by 6 x is equal to sine pi by 6 which is 1 by 2 which is 0.5. Then for y is equal to sine pi by 4 which is 1 by root 2 which is 0.71. y is equal to pi by 3 x is equal to pi by 3 which is root 3 by 2 which is 0.87 is equal to pi by 2 x is equal to sine pi by 2 which is 1. Now for drawing the graph of the inverse trigonometric function y is equal to sine inverse x we will plot these points. Now by taking 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 drawn a graph of the point minus 1 minus pi by 2 on the graph minus 0.87 and minus pi by 3 on the graph. Now this is the required point on the graph. Similarly we will plot all the other points on the graph. So we have plotted all the points on the graph and by drawing all these points we are getting the graph of the inverse trigonometric function y is equal to sine inverse x. Now here we can observe that y increases monotonically from minus pi by 2 to pi by 2 as x increases from minus 1 to 1. And also if 0 then sine inverse of 1 is pi by 2 so minus 1 is minus pi by 2. First trigonometric function sine inverse x minus 1 is the graph of cosine inverse x. Now let us define a function f from the closed interval minus 1 to 1 to the closed interval 0 to pi by f of x is equal to plus. We have restricted the domain of cos x to the closed interval 0 to pi that domain the closed interval minus 1 to 1 is equal to minus 1 and x is less than equal to 1. So we can say that cosine inverse x is defined to minus 1 to 1 only. And the range is the closed interval 0 to pi that is the possible f of x values for the graph of cosine inverse x will be less than equal to pi is equal to cosine inverse x that is let y is equal to cosine inverse x. Now let us draw the graph of y is equal to cosine inverse x. Now y is equal to cosine inverse x implies x is equal to now for the different values of x and y we will draw now for y is equal to cos pi which is minus 1. For y is equal to x is equal to cos of pi pi by 6 is equal to minus root 3 by 2 which is equal to minus 0.87. For y is equal to pi by 4 which is 1 by root 2 which is minus 0.71. Now for y is equal to 2 pi by 3 2 pi by 3 which is equal to minus 1 by 2 which is minus 0.5. Now for y is equal to pi by 2 x is equal to cos pi by 2 which is 0. And for y is equal to 0.3 x is equal to cos of pi by 3 which is 1 by 2 which is 0.5. Now for y is equal to x is equal to cos pi by 4 which is 1 by root 2 which is 0.71. And for y is equal to pi by 6 x is equal to cos of pi by 6 which is root 3 by 2 which is 0.87. And for y is equal to 0 x is equal to cos 0 which is 1. Now for drawing the graph of the inverse symmetric function y is equal to cos of x. We will plot these points. On 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. And then we have drawn a graph that we will plot on the graph minus 1. So this is the required point on the graph. We will plot all the other points 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 the inverse symmetric function y is equal to cos inverse n. Now here we can observe that y decreases momentarily from pi to 0 increases from minus 1 also as inverse of minus 1 of 1 is 0. In the closed interval minus 1 to 1 only. So in this session we have learnt about the graphs of the inverse symmetric functions y is equal to sin inverse x and y is equal to cos inverse x. So this will be the session where we will have enjoyed the session.