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.):
- The line is a straight line through the point.
- The curve goes through that point.
- The curve is differentiable in a neighborhood of the point.
- 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 
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
- Tangent in On Line Etymological Dictionary
- Tangent in MathWorld.
- Tangent in Wikipedia.
- History of Trigonometry in Wikipedia
- Earliest known uses of some of the words of mathematics
- Definitions in abstractmath.org.
- Images and Metaphors in abstractmath.org
- Tangent Line.nb Mathematica Notebook containing the code for the graphs in this article.
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