Why do "secant" lines and "1/cos" share a name?

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Coding
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Why do "secant" lines and "1/cos" share a name?

Postby Coding » Mon Sep 19, 2011 9:05 pm UTC

I asked my calculus instructor this question, and she was unsure.

Can anyone enlighten me on the relationship (if any) between the trig functions like secant and tangent, and their correspondingly named lines in geometry / calculus?

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Re: Why do "secant" lines and "1/cos" share a name?

Postby Aiwendil » Mon Sep 19, 2011 11:32 pm UTC

The word "secant" comes from Latin "secare", "to cut". The meaning in reference to the "secant line" that crosses a given curve is pretty straightforward, then; the secant is thought of as cutting across the curve.

The trigonometric meaning of "secant" apparently comes from another meaning of "secant line" - to wit, the line segment that starts at the center of a circular arc, passes through one end of the arc, and terminates when it intersects the the line tangent to the other end of the arc (thus cutting across the arc itself, I suppose). The ratio of this length to the radius is what we call a secant today.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby gmalivuk » Mon Sep 19, 2011 11:56 pm UTC

It might help to see a geometric picture of the secant and other not-sine-or-cosine trig functions:

Image

So the tangent, for example, is the distance along the line *tangent* to the circle along the x-axis at which the angle intersects it.
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Re: Why do "secant" lines and "1/cos" share a name?

Postby Coding » Tue Sep 20, 2011 12:09 am UTC

Interesting. Thanks!

I don't think I had ever used secants or cosecants of circles in any substantive way, so I had never thought about them beyond just translating them into "1 / cos" and "1 / sin" for homework and tests. Never learned to visualize them like cosine, sine, and tangent, anyway.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby Tirian » Tue Sep 20, 2011 12:51 am UTC

After seeing that excellent diagram*, the only question that's left is why we name the third sort of function "sine" instead of, I don't know, "semi-chord". Turns out to be an interesting story, because the Babylonians and Arabs did exactly that, and in the twelfth century someone was translating an Arab document and misread "jiba" (meaning chord) as "jaib" (meaning bosom) and translated it as the Latin word for chest, which is "sine". At least, that's the legend.

* In retrospect, that diagram is only partially excellent. It is wrong in making "x" pull double-duty as the angle and the x-coordinate of the point where the ray hits the unit circle.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby jestingrabbit » Tue Sep 20, 2011 4:25 am UTC

trigo_functions.gif
I like big triangles.
trigo_functions.gif (6.68 KiB) Viewed 5934 times
Same diagram, rearranged a little bit.
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Re: Why do "secant" lines and "1/cos" share a name?

Postby gmalivuk » Tue Sep 20, 2011 3:22 pm UTC

That makes the reason for calling it "tangent" even more clear (which probably means that is closer to the diagram people had in mind when naming this stuff in the first place). It also illustrates why the functions and their cofunctions go together, as all the "regular" functions are lines below the angle, while the cofunctions are the lines on the other side.
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Re: Why do "secant" lines and "1/cos" share a name?

Postby Jyrki » Wed Sep 21, 2011 6:08 pm UTC

This thread gives me a reason to bring something up. You know. I never use either sec x or csc x. I always use 1/cos x or 1/sin x. This is the way it was taught to me as an undergrad, and now that I'm teaching first year calculus I won't even mention secant or cosecant to my students. Today I discussed this with colleagues, and we all agreed that we have only seen secant or cosecant in U.S. based calculus books and Mathematica outputs. Admittedly we hardly see calculus textbooks that are not U.S. based, so we can't read too much into that :)

Ok, to each their own. As long as we know what we mean, we can communicate with each other so no harm. But this is unlikely to be just a Finnish thing. We are usually too conformist to rock the boat like that. We must have copied it from either Germany (or Sweden which indirectly is likely to come to much the same thing).

So finally comes my question. Is this a question of geography in that in some countries they use all six trig function, but some other countries only use four (or three as cotangent definitely gets less attention). Has this changed over the course of history? I would like to survey this. What is your personal view? Do you use either {1/cos x, 1/sin x} or {sec x, csc x} exclusively? Do you use both? Do you care? Where are you from? Is it possible to poll this here? A simple poll won't do, because I am interested in finding out, whether there is regional difference.

My calculator (when I still owned one a couple of decades ago) only had sin, cos and tan keys. We can't read too much into that either, because those are surely the three most common trig functions. And the calculator did have a 1/x key also :)

This was not meant to be a flame-bait or an opening salvo in yet another pointless Transatlantic pissing contest. I truly am curious about this. I do realize that my anti-tau rant in another thread was based on the need to minimize the number of divisions (as they take up more space), so my view here is not really aligned with my view on pi vs. tau :)

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Re: Why do "secant" lines and "1/cos" share a name?

Postby gorcee » Wed Sep 21, 2011 6:15 pm UTC

Gmalivuk's diagram is very illustrative. As others said, we have those terms because they all relate to different quantities.

It's just a mathematical consequence that we have the relationships we do (ie, sec = 1/cos). And because these relationships are so straightforward and simple, we really only care about the three main ones, which is why you see those on a calculator (also, limited buttons!).

Occasionally, it might be easier to remember the integral rule for secant formulae than 1/cos formulae, so it's certainly worth recognizing those functions and having them around.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby letterX » Wed Sep 21, 2011 6:49 pm UTC

Also, the large number of names for various functions of sin, cos and tan is largely historical. Back when trig functions were evaluated by looking them up in a big table 'o values, it was much faster and more accurate to find the value for sec(x) than to find the value of cos(x) and take its reciprocal. There are even more names for functions we tend not to think about anymore, like the 'haversine', or sin^2(x/2). Turns out that when you're doing something like computing distance along great circles over and over again, it's more helpful to have a table of haversines. So you give it a new name, pay a bunch of people to calculate all the values, and print up a handy new book.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby Tirian » Wed Sep 21, 2011 7:14 pm UTC

Also, the notion that there are six primitive trigonometric functions is a geometric-centric approach. Pick up a real analysis book and they'll define sine and cosine as the only elementary functions, since they are the decomposition of f(x)=eix into real and imaginary parts.

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Re: Why do "secant" lines and "1/cos" share a name?

Postby skullturf » Wed Sep 21, 2011 9:06 pm UTC

Jyrki wrote:So finally comes my question. Is this a question of geography in that in some countries they use all six trig function, but some other countries only use four (or three as cotangent definitely gets less attention). Has this changed over the course of history? I would like to survey this. What is your personal view? Do you use either {1/cos x, 1/sin x} or {sec x, csc x} exclusively? Do you use both? Do you care? Where are you from? Is it possible to poll this here? A simple poll won't do, because I am interested in finding out, whether there is regional difference.


I went to secondary school and university in Canada, and have taught calculus for about a decade at several universities in Canada and in the United States.

In the calculus textbooks I've used, all six trig functions tend to be used. However, cosecant and cotangent are probably used less often.

I often tell my students that one doesn't see cosecant or cotangent very frequently, and that one can get away with thinking of {sin, cos, tan, sec} as the four "main" trig functions.

I do find that for remembering derivatives and antiderivatives of common functions, it's useful to think of sec as being sec, as opposed to 1/cos. (But I may be biased by the simple fact that I'm accustomed to the "sec" notation.)


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