Tag Archives: secant

exposition, math, understanding math

Liberal-artsy people

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I graduated from Oberlin College with a B.A. as a math major and minors in philosophy and English literature, with only three semesters of science courses.  I was and am "liberal-artsy".   As professor of math at Case Western Reserve University,  I had lots of colleagues in both pure and applied math who started out with B.Sc. degrees. We did not always understand each other very well!

Caveat: "Liberal-artsy" and "Narrowly Focused B.Sc. type" (I need a better name) are characteristics that people may have in varying amounts, and many professors in science and math have both characteristics.   I do, myself, although I am more L.A. that B.Sc.  Furthermore, I know nothing about any sociological or cognitive-science research on these characteristics.  I am making it all up as I write.  (This is a blog post, not a tome.)

I recently posted on secants and  tangents.  These articles were deliberately aimed to tickle the interests of L.A.  students.

Liberal-artsy types want to know about connections between concepts.  In each post, I wrote on both common meanings of the words (secant line and function, tangent line and function) and the close connections between them.  Some trig teachers / trig texts tell students about these connections but too many don't.   On the other hand, many B.Sc. types are left cold by such discussions.  B.Sc. types are goal-oriented and want to know a) how do I use it? b) how do I calculate it?  They get impatient when you talk about anything else.  I say point out these connections anyway.

L.A. types want to know about the reason for the name of a concept.  The post on secants refers to the metaphor that "secant" means "cutting". This is based on the etymology of "secant", which is hidden to many students  because it is based on Latin.  The post makes the connection that the "original" definition of "secant" was the length of a certain line segment generated by an angle in the unit circle. The post on tangents makes an analogous connection, and also points out that most tangent lines that students see touch the curve at only a single point, which is not a connotation of the English word "touch".

Many people think they have learned something when they know the etymology of a word.  In fact, the etymology of a word may have little or nothing to do with its current meaning, which may have developed over many centuries of metaphors that become dead, generate new metaphors that become dead, umpteen times, so that the original meaning is lost.  (The word "testimony" cam from a Latin phrase meaning hold your testicles, which is really not related to its meaning in present-day English.)

So I am not convinced that etymologies of names can help much in most cases.  In particular, different mathematical definitions of the same concept can be practically disjoint in terms of the data they use, and there is no one "correct" definition, although there may be only one that motivates the name.  (There often isn't a definition that motivates the name.  Think "group".)  But I do know that when I mention the history of a name of a concept in class, some students are fascinated and ask me questions about it.

L.A. types are often fascinated by ETBell-like stories about the mathematician who came up with a concept, and sometimes the stories illuminate the mathematical idea.  But L. A. types often are interested anyway.  It's funny when you talk about such a thing in class, because some students visibly tune out while others noticeably perk up and start paying attention.

So who should you cater to?  Answer:  Both kinds of students.  (Tell interesting stories, but quickly and in an offhand way.)

The posts on secants and tangents also experimented with using manipulable diagrams to illustrate the ideas.  I expect to write about that more in another post.

For more about the role of definitions, check out the abmath article and also Timothy Gowers' post on definitions (one of a series of excellent posts on working with abstract math).


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exposition, language, math, Uncategorized, understanding math

Tangents

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code is the Mathematica Notebook Tangent Line.nb, which is available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

This is an experiment in exposition of the mathematical concepts of tangent.  It follows the same pattern as my previous post on secant, although that post has explanations of my motivation for this kind of presentation that are not repeated here.

Tangent line

A line is tangent to a curve (in the plane) at a given point if all the following conditions hold (Wikipedia has more detail.):

  1. The line is a straight line through the point.
  2. The curve goes through that point.
  3. The curve is differentiable in a neighborhood of the point.
  4. The slope of the straight line is the same as the derivative of the curve at that point.

In this picture the curve is $ y=x^3-x$ and the tangent is shown in red. You can click on the + signs for additional controls and information.

Etymology and metaphor

The word “tangent” comes from the Latin word for “touching”. (See Note below.) The early scholars who talked about “tangent” all read Latin and knew that the word meant touching, so the metaphor was alive to them.

The mathematical meaning of “tangent” requires that the tangent line have slope equal to the derivative of the curve at the point of contact. All of the red lines in the picture below touch the curve at the point (0, 1.5). None of them are tangent to the curve there because the curve has no derivative at the point:

The curve in this picture is defined by

The mathematical meaning restricts the metaphor. The red lines you can generate in the graph all touch the curve at one point, in fact at exactly at one point (because I made the limits on the slider -1 and 1), but there are not tangent to the curve.

Tangents can hug!

On the other hand, “touching” in English usage includes maintaining contact on an interval (hugging!) as well as just one point, like this:

The blue curve in this graph is given by

The green curve is the derivative dy/dx. Notice that it has corners at the endpoints of the unit interval, so the blue curve has no second derivative there. (See my post Curvature).

Tangent lines in calculus usually touch at the point of tangency and not nearby (although it can cross the curve somewhere else). But the red line above is nevertheless tangent to the curve at every point on the curve defined on the unit interval, according to the definition of tangent. It hugs the curve at the straight part.

The calculus-book behavior of tangent line touching at only one point comes about because functions in calculus books are always analytic, and two analytic curves cannot agree on an open set without being the same curve.

The blue curve above is not analytic; it is not even smooth, because its second derivative is broken at $x=0$ and $x=1$. With bump functions you can get pictures like that with a smooth function, but I am too lazy to do it.

Tangent on the unit circle

In trigonometry, the value of the tangent function at an angle $ \theta$ erected on the x-axis is the length of the segment of the tangent at (1,0) to the unit circle (in the sense defined above) measured from the x-axis to the tangent’s intersection with the secant line given by the angle. The tangent line segment is the red line in this picture:


This defines the tangent function for $ -\frac{\pi}{2} < x < \frac{\pi}{2}$.

The tangent function in calculus

That is not the way the tangent function is usually defined in calculus. It is given by \tan\theta=\frac{\sin\theta}{\cos\theta}, which is easily seen by similar triangles to be the same on -\frac{\pi}{2} < x < \frac{\pi}{2}.

We can now see the relationship between the geometric and the $ \frac{\sin\theta}{\cos\theta}$ definition of the tangent function using this graph:


The red segment and the green segment are always the same length.
It might make sense to extend the geometric definition to $ \frac{\pi}{2} < x < \frac{3\pi}{2}$ by constructing the tangent line to the unit circle at (-1,0), but then the definition would not agree with the $ \frac{\sin\theta}{\cos\theta}$ definition.

References

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exposition, language, language of math, math, Mathematica, understanding math

Case Study in Exposition: Secant

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The interactive examples in this post require installing Wolfram CDF player, which is free and works on most desktop computers using Firefox, Safari and Internet Explorer, but not Chrome. The source code comes from several Mathematica notebooks lists in the References. The notebooks are available for free use under a Creative Commons Attribution-ShareAlike 2.5 License. The notebook can be read by CDF Player if you cannot make the embedded versions in this post work.

Pictures, metaphors and etymology

Math texts and too many math teachers do not provide enough pictures and metaphors to help students understand a concept.  I suspect that the etymology of the technical terms might also be useful. This post is an experimental exposition of the math concept of “secant” that use pictures, metaphors and etymology to describe the concept.

The exposition is interlarded with comments about what I am doing and why.  An exposition directly aimed at students would be slimmer — but some explanations of why you are doing such and such in an exposition are not necessarily out of place every time!

Secant Line

The word “secant” is used in various related ways in math.  To start with, a secant line on a curve is the unique line determined by two distinct points on the curve, like this:


The word “secant” comes from the Latin word for “cut”, which came from the Indo-European root “sek”, meaning “cut”.  The IE root also came directly into English via various Germanic sound changes to give us “saw” and “sedge”.

The picture

Showing pictures of mathematical objects that the reader can fiddle with may make it much easier to understand a new concept.  The static picture you get above by keeping your mitts off the sliders requires imagining similar lines going through other pairs of points. When you wiggle the picture you see similar lines going through other pairs of points.  You also get a very strong understanding of how the secant line is a function of the two given points.  I don’t think that is obvious to someone without some experience with such things.

This belief contains the hidden claim that individuals vary a lot on how they can see the possibilities in a still picture that stands as an example of a lot of similar mathematical objects.  (Math books are full of such pictures.)  So people who have not had much practice learning about possible variation in abstract structures by looking at one motionless one will benefit from using movable parametrized pictures of various kinds.  This is the sort of claim that is amenable to field testing.

The metaphor

Most metaphors are based on a physical phenomenon.  The mathematical meanings of “secant” use the metaphor of cutting.  When the word “secant” was first introduced by a European writer (see its etymology) in the 16th century, the word really was a metaphor.   In those days essentially every European scholar read Latin. To them “secant” would transparently mean “cutting”.  This is not transparent to many of us these days, so the metaphor may be hidden.

If you examine the metaphor you realize that (like all metaphors) it involves making some remarkably subtle connections in your brain.

  • The straight line does not really cut the curve.  Indeed, the curve itself is both an abstract object that is not physical, so can’t be cut, and also the picture you see on the screen, which is physical, but what would it mean to cut it?  Cut the screen?  The line can’t do that.
  • You can make up a story that (for example) the use was suggested by the mental image of a mark made by a knife edge crossing the plane at points a and b that looks like it is severing the curve.
  • The metaphor is restricted further by saying that it is determined by two points on the curve.   This restriction turns the general idea of secant line into a (not necessarily faithful!) two-parameter family of straight lines.  You could define such a family by using one point on the curve and a slope, for example.  This particular way of doing it with two points on the curve leads directly to the concept of tangent line as limit.

Secant on circle

Another use of the word “secant” is the red line in this picture:


This is the secant line on the unit circle determined by the origin and one point on the circle, with one difference: The secant of the angle is the line segment between the origin and the point on the curve.  This means it corresponds to a number, and that number is what we mean by “secant” in trigonometry.

To the ancient Greeks, a (positive) number was the length of a line segment.

The Definition

The secant of an angle $\theta$ is usually defined as $\frac{1}{\cos\theta}$, which you can see by similar triangles is the length of the red line in the picture above.

In many parts of the world, trig students don’t learn the word “secant”. They simply use $\frac{1}{\cos\theta}$.

This illustrates important facts about definitions:

  • Different equivalent definitions all make the same theorems true.
  • Different equivalent definitions can give you a very different understanding of the concept.

The red-line-segment-in-picture definition gives you a majorly important visual understanding of the concept of “secant”.  You can tell a lot from its behavior right off (it goes to infinity near $\pi/2$, for example).

The definition $\sec\theta=\frac{1}{\cos\theta}$ gives you a way of computing $\sec\theta$.  It also reduces the definition of $ \sec\theta$ to a previously known concept.

It used to be common to give only the $ \frac{1}{\cos\theta}$ definition of secant, with no mention of the geometric idea behind it.  That is a crime.  Yes, I know many students don’t want to “understand” stuff, they only want to know how to do the problems.  Teachers need to talk them out of that attitude.  One way to do that in this case is to test them on the geometric definition.

Etymology

This idea was known to the Arabs, and brought into European view in the 16th century by Danish mathematician Thomas Fincke in “Geometria Rotundi” (1583), where the first known use of the word “secant” occurs.  I have not checked, but I suspect from the title of the book that the geometric definition was the one he used in the book.

It wold be interesting to know the original Arabic name for secant, and what physical metaphor it is based on.  A cursory search of the internet gave me the current name in Arabic for secant but nothing else.

Graph of the secant function

The familiar graph of the secant function can be seen as generated by the angle sweeping around the curve, as in the picture below. The two red line segments always have the same length.


References

Mathematica notebooks used in this post:

 

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