Here is one of the questions that I ask my class of future secondary mathematics teachers to answer.
A student asks, “My father was helping me with my homework last night and he said the book is wrong. He said that and , because and . But the book says . He wants to know why we are using a book that has mistakes.”
This is a very similar question to the simplification of , which was discussed in yesterday’s post. My experience has been that the above misconceptions involves confusion surrounding two very similar-sounding questions.
Question #1: Find all values of so that .
Question #2: Find the nonnegative value of so that .
The first question can be restated as solving , or finding all roots of a second-order polynomial. Accordingly, there are two answers. (Of course, the answers are and , written more succinctly as .) The second question asks for the positive answer to Question #1. This positive answer is defined to be , or .
In other words, it’s important to be sure that you’re answering the right question.
Is it all that important that is chosen to be the nonnegative solution to Question #1? Not really. We could have easily chosen the negative answer. The reason we choose the positive answer and not the negative answer can be answered in one word: tradition.
We want to be a well-defined function that produces only one output value, and there’s no mathematically advantageous reason for choosing the nonnegative answer aside from the important consideration that everyone else does it that way. And though students probably won’t remember this tidbit of wisdom when the time comes, the same logic applies when choosing the range of the inverse trigonometric functions.
Of course, for the present case, it totally makes sense to take the positive, less complicated answer as the output of . However, this won’t be as readily apparent when we consider the inverse trigonometric functions.