Inverse Trig Functions - Cornell University（逆三角函数康奈尔大学）.pdf

Inverse Trig Functions c A Math Support Center Capsule February 12, 2009 Introduction Just as trig functions arise in many applications, so do the inverse trig functions. What may be most surprising is that they are useful not only in the calculation of angles given the lengths of the sides of a right triangle, but they also give us solutions to some common integrals. For example, suppose you need to evaluate the following integral: b √ 1 2 dx a 1 − x for some appropriate values of a and b. You can use the inverse sine function to solve it! In this capsule we do not attempt to derive the formulas that we will use; you should look at your textbook for derivations and complete explanations. This material will simply summarize the key results and go through some examples of how to use them. As usual, all angles used here are in radians. Restrictions on the Domains of the Trig Functions A function must be one-to-one for it to have an inverse. As we are sure you know, the trig functions are not one-to-one and in fact they are periodic (i.e. their values repeat themselves periodically). So in order to define inverse functions we need to restrict the domain of each trig function to a region in which it is one-to-one but also attains all of its values. We do this by selecting a specific period for each function and using this as a domain on which an inverse can be defined. Clearly there are an infinite number of different restrictions we could chose but the following are choices that are normally used. 1 Standard Restricted D