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Unformatted text preview: Chapter 11 Inverse Trigonometric functions In this chapter, we investigate inverse trigonometric functions 1 . As in other examples, the inverse of a given function leads to exchange of the roles of the dependent and independent variables, as well as the the roles of the domain and range. Geometrically, an inverse function is obtained by reflecting the original function about the line y = x . However, we must take care that the resulting graph represents a true function, i.e. satisfies all the properties required of a function. The domains of sin( x ) and cos( x ) are both-∞ < x < ∞ while their ranges are- 1 ≤ y ≤ 1. In the case of the function tan( x ), the domain excludes values ± π/ 2 as well as angles 2 nπ ± π/ 2 at which the function is undefined. The range of tan( x ) is-∞ < y < ∞ . There is one difficulty in defining inverses for trigonometric functions: the fact that these func- tions repeat their values in a cyclic pattern means that a given y value is obtained from many possible values of x . For example, all of the values x = π/ 2 , 5 π/ 2 , 7 π/ 2, etc all have identical sine values sin( x ) = 1. We say that these functions are not one-to -one . Geometrically, this is just saying that the graphs of the trig functions intersect a horizontal line in numerous places. When these graphs are reflected about the line y = x , they would intersect a vertical line in many places, and would fail to be functions: the function would have multiple y values corresponding to the same value of x , which is not allowed. The reader may recall that a similar difficulty was encountered in an earlier chapter with the inverse function for y = x 2 . We can avoid this difficulty by restricting the domains of the trigonometric functions to a portion of their graphs that does not repeat. To do so, we select an interval over which the given trigonometric function is one-to -one, i.e. over which there is a unique correspondence between values of x and values of y . (This just mean that we keep a portion of the graph of the function in which the y values are not repeated.) We then define the corresponding inverse function, as described below. Arcsine is the inverse of sine The function y = sin( x ) is one-to-one on the interval- π/ 2 < x < π/ 2. We will define the associated function y = Sin ( x ) (shown in red on Figures 11.2, 11.1 by restricting the domain of the 1 The main importance of this material is that it introduces functions like Arctan that play a significant role in integral calculus next semester. However, it is possible to abbreviate this material, and make it part of the discussion of Chapter 10 to save time for approximation methods and differential equations at the end of this course. The example of the visual response in Sec 11.2 can also be handled by using implicit differentiation of the trig functions, rather than by setting up a derivative of the inverse trigonometric functions....
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This note was uploaded on 05/19/2011 for the course MATH 102 Math 102 taught by Professor Allard during the Winter '09 term at UBC.
- Winter '09