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Re: Why trig?
Posted:
Apr 2, 1997 3:02 PM
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On Wed, 2 Apr 1997, John Sheehan wrote:
> Discuss what sorts of issues came up with the algebraic functions. Remember
> the square root? It lead to irrational numbers. So do trig functions.
So do linear functions, you know. I guess you mean that the
square root of an *integer* can be irrational. On the other hand, the
cosine of (2*pi) is 1, very rational. I think you will have to phrase
your discussion with care.
> They
> are part of a large family called transcendental functions.
True, but to prove that "sine" is not an algebraic function, for example,
is not for a high school course. On the other hand, any student of
trigonometry ought to be able to understand that cos(arctan x) =
1/(sqrt(1+x^2)), an algebraic function defined for all real x. Most
calculus students have trouble with this one, because it turns up too
suddenly, in connection with derivatives.
> Throw in a little history.
Now you're talking! To carry back the formula for cos(A+B) to ancient
times is, however, not easy. There is a proof of the equivalent
attributed to Ptolemy, but it requires quite a bit of knowledge of Euclid
to follow it. But even if a trig class doesn't get the full story, it is
very good to tell as much of the chronological story as possible. On the
other hand, historical trigonometry is not about periodic functions, but
is confined to angles inscribed in circles somewhere, i.e. angles that
can occur in triangles; and the extension of the trig functions to larger
domains is more recent. Here, history can be mischievous if one is not
careful. Old fashioned trig books would give a definition of sine as a
ratio in a right triangle, hence for acute angles only, and then extend
the definition to the other quadrants by a woefully long list of
adjustments. From a strictly historical point of view that may be more
or less the way it happened, but it is better to give the y/r definition
as soon as possible, and *use* it, either in talking about complex
numbers or about polar coordinates in general. And then *use* complex
numbers for something as soon as possible, e.g. De Moivre's theorem for
roots of unity, or any other complex number, according to how much time
your curriculum gives you for such impractical things.
Mention some (diverse) applications. Tie it in to
> what they already know. Give an idea about where this course leads to. That
> to me is "motivation."
>
Here is some motivation for trig functions, apart from solving
triangles, and apart from more sophisticated applications such as
alternating current and Fourier Series descriptions of musical tones:
Roots: How come there are two square roots of a positive number
but only one cube root? How come there are *no* square roots of a
negative number, but one cube root? What about fourth roots? nth roots?
Directions: In Manhattan one tells a stranger how to get
somewhere by saying it is five blocks north and two blocks west from
where you are. In an Iowa cornfield, what do you tell him? And then,
when the building developer spoils the landscape with housing or office
buildings, how do you change the Iowa description back to the New York style?
Graphs: Explain the cardioid microphone, and why it is that
people can see so much of the TV screen from even fairly extreme
positions to the right and left of the selfish people who had turned the
set on for themselves only.
> Then, roll up your sleeves and teach them some trig.
>
Right on.
Ralph A. Raimi Tel. 716 275 4429, or (home) 716 244 9368
University of Rochester Fax 716 244 6631
Rochester, NY 14627 Homepage: http://www.math.rochester.edu/u/rarm
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