Defining the Complement of an AngleDate: 08/04/2007 at 19:29:09 From: Steve Subject: What is a rigorous definition of an angle's complement? It's true that sin(x°) = cos(90° - x°) for all real x. So, for any angle x°, is 90° - x° defined as its complement? Wolfram says x°'s complement is 90° - x°, with no restrictions. Blitzer, Precalculus, Prentice Hall, 2004, says complements (and supplements) must be positive, so some angles don't have complements. His example: 123° doesn't have a complement b/c 90 - 123 < 0. Yet sin(123°) = cos(90° - 123°). So in practice we say the sine of an angle is equal to the cosine of the angle's complement, and obviously that can be negative. I'm teaching Trig and Analytic Geo for the first time this Fall, and I'm starting to prepare now, and trying to get all my facts straight. Date: 08/04/2007 at 23:11:30 From: Doctor Peterson Subject: Re: What is a rigorous definition of an angle's complement? Hi, Steve. It all depends on context and perspective. Different authors in different texts have different needs, so they make different restrictions. Even within a single text there can be several contexts which require different perspectives on the definition of words. In geometry, an angle tends to be seen as a figure, and its measure just has to describe the relation between the two rays that make up that figure; so it is always between 0 and 180 degrees. Since the complement of an angle also has to be an angle, you can only take the complement of an acute angle in this sense. But in trigonometry, an angle is commonly thought of instead as a rotation--an action that will result in a given figure, not the figure itself. In this sense, an angle can be positive or negative, and can be greater than 360 degrees. All real numbers correspond to angles. When you treat angles this way, there is no reason not to allow complements to apply to all real numbers as well. I'm not sure why a precalc book, which presumably has an analytical focus, would restrict the usage of complements; I suppose that must come up when they are looking at geometrical problems, not at trig in its broader, analytical form. In teaching from such a book, I would downplay the restriction and apply it only where it makes sense, being sure to drop the restriction when talking about identities such as those you are referring to. The main idea of a complement is 90-x; restrictions are not an essential part of the definition. In fact, that might be a good general rule: although sometimes mathematicians tend to give a lot of attention to restrictions (since they are what allow you to state theorems precisely), in initial teaching we need to focus on the positive-- what something means in its broadest sense, rather than on the special cases where it doesn't apply. If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 08/05/2007 at 07:20:47 From: Steve Subject: Thank you (What is a rigorous definition of an angle's complement? ) Thank you, Doctor Peterson! - Steve |
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