Wikipedia:Reference desk/Archives/Mathematics/2015 July 18

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

July 18

Ordering the trigonometric functions

I've always thought the most natural order was "sine, cosine, tangent, cotangent, secant, cosecant". But for some reason, the most common order for textbooks to mention them is "sine, cosine, tangent, cosecant, secant, cotangent". Any thoughts?? Georgia guy (talk) 00:40, 18 July 2015 (UTC)

Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. GeoffreyT2000 (talk) 03:58, 18 July 2015 (UTC)

The idea that there is a "natural order" is ridiculous. (The fact that most of these archaic names still exist and are used by anyone is also ridiculous.) --JBL (talk) 13:35, 18 July 2015 (UTC)

How are these names archaic?? Georgia guy (talk) 13:42, 18 July 2015 (UTC)

The name "sine" is from an accidental mistranslation in the Middle Ages. There is no particular need to have separate names for most of these functions, given the simple relations between them. We've collectively stopped using the versine and haversine, and good riddance. Cotangent and cosecant and probably secant (though I have a personal fondness for the latter) deserve the same fate. --JBL (talk) 18:14, 18 July 2015 (UTC)

But if we stopped forcing Freshmen to memorize the derivative of arccosecant (and the hyperbolic version) then we might have to teach them something useful instead. Then where would we be? --RDBury (talk) 19:30, 18 July 2015 (UTC)

That one has a nice small wrinkle, though I'm not sure I ever retained it (the wrinkle or the derivative) in memory for more than a few seconds at best! The derivative of arccosecant is $-{\frac {1}{x^{2}{\sqrt {1-{\frac {1}{x^{2}}}}}}}$: and you can't simplify it to $-{\frac {1}{x{\sqrt {x^{2}-1}}}}$ because that messes up the sign for negative x. (For the derivative of hyperbolic arccosecant, replace the minus sign in the square root with a plus sign.) Double sharp (talk) 16:56, 20 July 2015 (UTC)

• I notice that Python's standard math library does not include cot, sec or csc. —Tamfang (talk) 07:33, 22 July 2015 (UTC)

Trigonometric functions, Wikipedia's article on the subject, lists sin, cos and tan in the first section, and then the reciprocals in the second section; presumably because that is the order they are usually presented to classes of teenagers, and also the order they are presented in introductory textbooks. Wikipedia is not a textbook and I think we should present the three fundamental ratios first - sin, tan and sec although not necessarily in that order. Secondly, we should present cos, cot and cosec - the three ratios based on the complementary angles. Dolphin (t) 12:32, 22 July 2015 (UTC)

I don't have any idea what "fundamental ratios" means, but also it seems like your comment should be on the article talk page, not here. --JBL (talk) 02:09, 23 July 2015 (UTC)

Perhaps it would be clearer if I called them the ratios with the fundamental names. Sine, secant and tangent are names that are used consistently elsewhere in mathematics. Conversely, the names cosine, cosecant and cotangent are not used elsewhere in mathematics - they are names generated by adding the prefix co to the three fundamental names. There are many examples similar to this in the English language; for example, we have the words clockwise and anticlockwise. The word clockwise is regarded as the fundamental whereas anticlockwise is a development of the fundamental. No author or mathematician would contemplate introducing the concept of the antilogarithm before the logarithm; or the antiderivative before the derivative. Dolphin (t) 06:07, 23 July 2015 (UTC)

I can't make much sense out of this, frankly. You say, Sine, secant and tangent are names that are used consistently elsewhere in mathematics. What does this mean? "Secant" and "tangent" are used elsewhere in mathematics, but with meanings that are almost unrelated to the trigonometric meaning (they're related etymologically, but not in any way that seems particularly significant mathematically). I am not aware of any non-trig mathematical meaning for "sine".

It is a perfectly defensible choice to introduce the definite integral before the derivative, and the exponential function before the logarithm.

Also, counterclockwise note to anyone who didn't know: Yanks call it counterclockwise; Brits, anticlockwise is conventionally the "positive" direction of rotation, so if you had to pick one to be primary in mathematics, I suppose you'd go with counterclockwise.

Anyway, this risks degenerating into a Scholastic argument and let's not go there. But the bottom line is that I don't think it's justifiable to present sine as "more fundamental" than cosine. It's an accident of history that cosine got the co-, not something with mathematical significance. --Trovatore (talk) 21:30, 23 July 2015 (UTC)

I think that's Dolphin's point - it's an accident of history, and we're stuck with it. We can introduce $\int _{0}^{x}f(t)\ dt$ before introducing the derivative, but we can't call it an antiderivative of $f(x)$ before we introduce the derivative. The only difference is that the antilogarithm and antiderivative actually have other names - cosine, not so much. -- Meni Rosenfeld (talk) 07:58, 24 July 2015 (UTC)

Hmm? I don't see any impediment whatsoever to introducing cosine before sine. It's just a name. Neither one is particularly meaningful to the student who hasn't seen them before, and it's not even obvious at first glance that co- is a prefix at all here.

In SI, the fundamental unit of mass is the kilogram, and that doesn't seem to cause any particular difficulty. The older metric engineering units (cgs system) used gram as the unit of mass, but it used the centimeter as the unit of length, so not all that different.

I think it would be frankly silly to choose the order in which we treat these functions based on their names. There's not much about them that's less important than the names. --Trovatore (talk) 08:32, 24 July 2015 (UTC)

I agree emphatically with Trovatore. --JBL (talk) 11:12, 24 July 2015 (UTC)

It could be argued that the functions are listed in decreasing order of usefulness. You need to have either sin or cos, and sin is the "fundamental" one. cos can be found by complementing angle, but is still very useful to get a concise representation. tan can be found by dividing sin and cos, but is useful to have on its own. The others are pretty much useless. Except for haversine, maybe I'll start using that :) -- Meni Rosenfeld (talk) 20:47, 23 July 2015 (UTC)

Sine is the fundamental one? Who found that out? Personally I think if we were starting from scratch we would take cosine to be fundamental and sine to be derived. It's a close call, though, and I don't claim there's any way of "proving" it one way or the other. --Trovatore (talk) 21:00, 23 July 2015 (UTC)

I was using Dolphin's terminology - regardless of which is objectively more fundamental, historically sin was given a fundamental name, and cos was given a name derived from it. So unless we want to actually change the names to different names, we're stuck with listing sin first (in absence of a compelling reason to list cos first). -- Meni Rosenfeld (talk) 07:45, 24 July 2015 (UTC)

I disagree, as I explain at more length above. --Trovatore (talk) 08:33, 24 July 2015 (UTC)

Historically the more fundamental functions are the chord (=2 sin(v/2)) and the arrow (=1–cos(v/2)), where v is the arc. See Ptolemy's theorem. Bo Jacoby (talk) 23:37, 23 July 2015 (UTC).

Just to play the devil's advocate, as a geometrical construction, I would say the tangent is more natural. Given an arc on the circle, the secant line subtended by the angle intersects the line at infinity in the tangent of the arc. This is a projective invariant of the arc, in contrast to the sine and cosine which requires the Euclidean structure to be used. To construct the sine and cosine, one needs to use a special rectifying coordinate system. Although these rectifying coordinates can canonically be constructed out of an arc and circle using Euclidean geometry, anything requiring special coordinates does not really "feel" as natural as coordinate invariant constructions like the tangent. Sławomir Biały (talk) 14:22, 24 July 2015 (UTC)

But that's the tangent line, not the tangent function. To recover the tangent function, it seems to me, you're back to coordinate specificity. Or am I missing your point? --Trovatore (talk) 21:22, 24 July 2015 (UTC)

No, I mean the tangent function. Take the secant line subtended by the arc of a circle. Intersect with the line at infinity. That point is the tangent function of the original arc. Here the tangent function takes values that are not numbers, but rather in an abstract real projective line. The conventional tangent function is given by the value of a projective parameter at the point of intersection. Sławomir Biały (talk) 21:49, 24 July 2015 (UTC)

Solid of revolution from parametric equations

Our article on Green's theorem gives three formulas for the area of a region bounded by a parametrically defined curve:

$A=\oint _{C}x\,dy=-\oint _{C}y\,dx={\tfrac {1}{2}}\oint _{C}(-y\,dx+x\,dy).$

(It is assumed here that C is oriented positively (counterclockwise), otherwise change signs.) The corresponding formulas for the volume of a solid of revolution when the region is revolved about the x-axis are:

$V=2\pi \oint _{C}xy\,dy=-\pi \oint _{C}y^{2}\,dx={\frac {2\pi }{3}}\oint _{C}y(-y\,dx+x\,dy).$

(It is also assumed here that the region lies above the x-axis, otherwise change signs, possibly again). These also follow easily from Green's theorem but could also be derived with more elementary methods. It seems like these would all be fairly standard, but the textbooks I've seen only seem to cover the second form. Has anyone seen the first and/or third form in print and if so where? --RDBury (talk) 08:24, 18 July 2015 (UTC)

The first form just computes the volume by cylindrical shells and the second form is by washers. In most textbooks, these topics are covered before line integrals, so it might be hard to find them written exactly in this way in print. For the last one, write $x=r\cos \theta$ and $y=r\sin \theta$, and this comes from a "standard" formula volume in spherical coordinates (here θ is the colatitude). Sławomir Biały (talk) 13:09, 18 July 2015 (UTC)

In the formulas for area, the first one computes it by thin horizontal rectangles, the second by thin vertical rectangles and the third by wedges with vertices at the origin. The corresponding volume formulas just revolve the rectangles or wedges about the x-axis. Which is what I meant by more elementary methods. In any case, I'm not looking for the why of the tradition but whether there are any nonconformists out there. --RDBury (talk) 17:40, 18 July 2015 (UTC)

Well, at least the first two formulas do appear, more or less exactly this way, in any calculus text that discusses volumes by slicing. They do omit the symbol $\oint$ in the description, but it is precisely the same integral, at least when the region of revolution is bounded between two graphs. Sławomir Biały (talk) 21:20, 18 July 2015 (UTC)