 Hello and welcome to the session. In this session, we will discuss domain and range of inverse trigonometric functions. Till now, we have discussed trigonometric functions, y is equal to sin x, y is equal to y is equal to terms. We also know the domain and range of these functions. Now we know that domain refers to the input values and range refers to the output values. Now here, domain of this function that is y is equal to sin x. The set of all real numbers and range is the closed interval minus 1 to 1 minus for the trigonometric function y is equal to cos x. Domain is the set of all real numbers and range is the closed interval minus 1 to 1. And for the trigonometric function y is equal to tan x, domain is the set of all real numbers except i will do it plus n power where n is any integer. And range is the set of all real numbers. Now we also know a function y is equal to f of x is 1, 1, 2 element in set capital X that is domain corresponds to a unique element in set capital Y that is the range. If a function is 1, 1 then we can find inverse of a function which is given by x is equal to f inverse of y. Now here for inverse function the domain is still y and range is set to between x. This means domain is a set of different values of y for which the function is defined and the range is the set of corresponding values of x for which the function is defined. Now in this session we will find inverse of trigonometric functions that is we will find inverse sine function, inverse cosine function and inverse tangent function, their graphs, domain and range. First of all let us discuss inverse sine function that is y is equal to sine inverse x. We will see that wherever the function y is equal to sin x is 1 to 1 in the domain of all real numbers. Now here let us consider the graph of y is equal to sine x. Now here the value of sine is equal to 1 by 2 is equal to 1 by 2. So here we have the orbit pairs 1 by 2. This means for two distinct values u of y it means it is not equal to 1. It does not have distinct values of y for distinct values of y is equal to sine x that is the function y is equal to sine x is not 1 to 1. So we cannot find the inverse of sine function its domain clear interval in this restricted domain has a distinct value of y. Then the function becomes 1 to 1 and we can find inverse sine function given y is equal to. Now we have an inverse percent sine is negative 0 to pi by 2. Simply one negative value minus pi by 2 is less than equal to x is less than 0. So exactly one negative value for in the closed interval minus pi by 2 to pi by 2. We have distinct value of y belonging to this interval this interval the values u to 1 in the closed interval minus pi by 2 to pi by 2 inverse exists in this interval. So here we have restricted domain that is where the domain is the closed interval minus pi by 2 to pi by 2. So the function is 1 to 1 now inverse sine function is given by now we know in inverse function the domain and range interchange. That is the domain of the given function becomes a range of its inverse range of the given function. Here comes a significant function y is equal to sine inverse x the domain the closed interval minus 1 to 1 and range is the closed interval minus pi by 2 to pi by 2. The domain is the closed interval minus 1 to 1. It means minus 1 is less than equal to x is less than equal to 1 that means here x varies from minus 1 to 1. The range is the closed interval minus pi by 2 to pi by 2. So is less than equal to y is less than equal to it means here y varies from minus pi by 2 to pi by 2. So here first time function with a restricted domain for the function y is equal to sine x. We have got the domain which is equal to the search containing the element x such that minus pi by 2 is less than equal to x is less than equal to such that minus 1 is less than equal to y is less than equal to 1. So this is the case for the sine function with a restricted domain. Now for the inverse sine function that is y is equal to domain is equal to minus 1 is less than equal to x is less than equal to 1. And range is such containing the element y such minus pi by 2 is less than equal to y is less than equal to pi by 2. It means to the closed interval minus 1 to 1 we get you should note that sine inverse of sine theta is equal to theta where theta belongs to the closed interval minus pi by 2 to 1 by 2. So y is equal to sine inverse of 1 by 2. Now we know that is 1 by 2 at an angle of pi by 6 that is is equal to 1 by 2. Therefore y is equal to sine inverse y is equal to pi by 6 inverse cosine function. Now the cosine function like sine function is not 1 to 1 its domain to make it 1 to 1 function that has an inverse function. Now let us discuss cosine function from this diagram. Now in first quadrant that is where x varies from 0 to pi by 2 the cosine function has exactly one non-negative value. Second quadrant where x from pi by 2 and less than equal to cosine function has exactly one negative value. As we know cosine function is positive in first quadrant and it is negative in second quadrant. So we have distinct value of y value of x belonging to the closed interval 0 to pi. So we restrict the domain of the function y is equal to cos x to the closed interval 0 to pi. So we define inverse cosine function given by y is equal to cos inverse of x with domain which is the closed interval minus 1 to 1 as the closed interval 0 to pi function y is equal to cos x has domain which is equal to less than equal to pi and range is a set containing the element y such that minus 1 is less than equal to y is less than equal to 1. For inverse cosine function that is y is equal to cos inverse of x is a set containing the element x such that minus 1 is less than equal to x is less than equal to 1 and range is a set containing the element y such that 0 is less than equal to y is less than equal to and here also you must note that cos inverse of cos theta is equal to theta. Theta belongs to the closed interval 0 to pi which we belong to principal value given by the closed interval 0 to pi. Now let us discuss inverse tangent function given by sine functions. The tangent function is also not 1 to 1 in its domain and inverse function. Now in the first quadrant that is for 0 is less than equal to x is less than pi by 2. There is exactly one long negative value of tangent in the fourth quadrant that is for minus pi by 2 is less than x is less than 0 with one negative value of tangent function to the open interval minus pi by 2 to pi by 2. Now here we have taken open interval because tan is not defined. We have distinct value of y for every value of x belonging to this open interval minus pi by 2 to pi by 2. So here we have restricted the domain to the open interval minus pi by 2 to pi by 2. Now tangent function given by y is equal to tan inverse of and domain of this function is set of real numbers and range is the open interval minus pi by 2 to pi by 2 and range is the tangent function with a restricted domain and here y is equal to tan x has domain which is equal to set containing the element x such that minus pi by 2 is less than x is less than pi by 2 and range is a set containing the element y such that y is a real number. Now tangent function that is y is equal to tan inverse x domain is a set containing the element x such that x is a real number and range is a set containing the element y such that minus pi by 2 is less than y is less than pi by 2. So you must note that tan inverse of tan theta is equal to theta where theta belongs to the open interval minus pi by 2 to pi by 2. So for the inverse trigonometric function y is equal to tan inverse x y which will belong to principal value given by the open interval minus pi by 2 So in this session we have learnt domain and range of inverse trigonometric functions and this completes our session. Hope you all have enjoyed the session.