Defining the Complement of an Angle

Date: 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