Talk:Trigonometric function

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Contents

1 "Jump start" 2 TODO List 3 Domain of cotan 4 Notation of Inverses 5 Etymology of Sine 6 About CSC (http://en.wikipedia.org/wiki/ComputerSciencesCorp) 7 Multiple of 3 deg 8 Image problem? 9 Other problems 10 Arabic words Jiba and Jaib (and TeX) 11 Query 12 "Generality"

"Jump start"

The "Jump start" section is...um...out of place, to say the least, and I don't see how it helps. May I suggest people spend some time to get a feel for the style of articles here before plunging headfirst? Revolver 07:45, 17 Sep 2004 (UTC)

First of all i have to admit it is incoherent with the otherwise strong technical representation of the article, and it certainly can in no way claim to be correct from a abstract mathematical attidude. However it shall help pupils (perhaps even some undergraduates) to get a better 'grip' of what angles are as well as trigonometric functions. I noticed that minors soon loose the connection to the simple explanation of the triangle subscribed in the unit circle and treat angles as if they were something highly abstract, thus often try to avoid them or misuse them although angles if seen from a simple point of view could be treated with the same simplicity as say the add operation of scalars (or something else which is mathematically banal). Perhaps we should make a own page out of this: 'angles in simple terms' or something like that. One thing is for sure however that it misfits here, and this will only be a temporary measure, as i too would dislike to see something such non-technical in a article where you expect the opposite. I intend to put some images to that sometime later. Hope my intentions became a bit clearer :) --Slicky 19:18, Sep 17, 2004 (UTC)

My primary concern had to do with style. The jump start section is written more in the style of a conversational undergraduate lecture transcribed to paper. While this may be good for an oral lecture, it doesn't translate well to an encyclopedia article, and it kind of goes against style standards of wikipedia. Moreover, the issues you raise about student understanding or interpretation seem to me to have more to do with angles, and perhaps the straightforward geometric treatment could be emphasised more there. As to your contention of the problem (e.g. calc students who seem to "forget" that trig functions have geometric interpretations), I certainly know what you're talking about. But it seems to me the best way to address this in articles is simply to give a strong presentation of the simpler geometric interpretation. And I think this is done well in the beginning of the article. I think it's the best we can do, is present information and make it accessible. Of course, to show HOW simple interpretation is useful falls under this. But, to explicitly point out this choice and drive home its merits, seems to me to reflect a pedagogical invasion. That part is really the teacher's responsibility, and if they don't do it, it's their fault, not ours. Revolver 07:52, 18 Sep 2004 (UTC)

Okay i outsourced this article now by creating a new one, titled as trigonometry in simple terms and put a link at the bottom beside the others. I never actually intended to mix those articles, however as you might have noticed i am relatively new as a wikipedian (although i am a joyful wikipedia user since years), and i fancy the idea of free information since well..... my earliest days. A coherent style and technically strong formulation is the key to gain a large audience, however the typical 'calc students' are left out and rapidly loose their interest in maths more and more, as their own mind-build universe of mathematical consistency crumbles more and more until they are seeking out for anything (subject-related) that avoids math as much as possible, without beeing aware that mathematics is the key to everything. Therefore i think it could help one or another to actually find back to the path of maths, not with the intention to make them fit enough to become a mathematical theorist but to at least apply it with grace and delight and make extensive use of it in technical studies/research. To put the emphasis of this rant into one sentence and conclude what i began to say: I deem it important that there are also articles that are less mathematical, less correct and provide much less information in much more words for those not-so inclined. (But frankly if i haven't misunderstood you entirely it seems to me as if you too second that, and i totally agree with you in that coherence in any article should be preserved to the utmost possible extend).--Slicky 18:13, Sep 18, 2004 (UTC)

You might want to see the articles trigonometry and applications of trigonometry. The latter esp. might overlap with some things you're trying to do. As for calc students tuning out math because it is presented with a coherent style and technically strong formulation...how else should it be presented? With a confused, jumbled style and technically weak and ambiguous formulation?? I'm sorry, I feel for students, but at some point they just have to accept the nature of math and the nature of studying math. Math is an exact science; if you can't handle its precision, then maybe it's not for you. I don't know what the "path of math" is. If it is motivation, certainly I agree, but I don't think this conflicts with a coherent style and technically strong formulation. The two approaches are not at war with each other. Ideally, even a theoretical presentation should be given with "grace and delight". I think this is an effect of the Bourbaki school that still affects advanced math textbooks. As for the one sentence conclusion: I do think what you're asking for is important, I just feel it belongs more at a place like Wikibooks, not an encyclopedia. Encyclopedia articles (IMO) should be reference tools and introductions to subjects, but not pedagogical tools. I do not expect to gain a true understand of molecular biology or biochemistry from wikipedia articles, e.g., but I do expect it to be invaluable reference and guide while in a class or self-study. Revolver 04:48, 19 Sep 2004 (UTC)

At further contemplation the only conclusion that i came up with is to agree with you, that wikipedia is not the right place for such over-simplified and thus partly incorrect (regarding the expressions) written articles/entries. (I actually thought how i would respond whilst i am seeking for something and end up reading some introductory stuff for minors wich lacks depth in every way, as i actually used wikipedia a lot for further research on topics that were not comprehensively enough covered in books or not at all.) So there definitely should be a clear boarderline between exactly in-depth entries and entries that are more personal and just excerpts (the latter one surely misfits for a place like wikipedia, except user pages of course ;) ). For now I'll take it off and perhaps reshape it into a somewhat better formulated and more comprehensive bookentry in wikipedia out of it. BTW: With 'The path of math' i just meant to have a certain fascination and respect for math, even if you just use it as a tool (applied maths for instance in exp. physics/physical chemistry,..), because we wouldn't be where we are without it. (oh and forgive my lazy upper-case placing) --Slicky 07:24, Sep 19, 2004 (UTC)

TODO List

Could someone proficient with Latex, edit all text-typed formulae and expressions into a Latex meta-description. That would improve the readability a lot and would ensure a better experience as we strive towarads browser-native MathML support. --Slicky 10:48, Sep 16, 2004 (UTC)

Domain of cotan

cotan(x)=cos(x)/sin(x)

cotan(x)=1/tan(x)

The first of these is defined where cos(x)=0; the second is not. Does the domain of cotan include values of x for which cos(x)=0 or not?

Brianjd 08:36, Sep 12, 2004 (UTC)

The short answer to your question is that the confusion evaporates if we follow the convention that 1/∞ = 0 and 1/0 = ∞. If that bothers you, then the answer is that the domain is the same, and the identity holds everywhere that both sides are defined. If you want to use cot(x) = 1/tan(x) as a definition, then you should do it piecewise, using this where cos(x) is not 0, and then "plugging in the right value" when it is. Revolver 07:52, 18 Sep 2004 (UTC)

Notation of Inverses

which notation is more common for inverses: arcsin or sin^-1 ? Which came first? -- Tarquin 12:00 Mar 6, 2003 (UTC)

The notation f^-1 always means the inverse of f, never the multiplicative inverse of f. In programming we obviously have to use arcsin; I don't know about other places.

Brianjd 08:33, Sep 12, 2004 (UTC)

Etymology of Sine

…the modern word "sine" comes from a mistranslation of the Hindu jiva.

That seems farfetched and thus potentially interesting—please tell us more! What does jiva mean in Hindu? What's your source on this? The standard etymology of English sine is derivation from Latin sinus [curve, bend], which is pretty suggestive of the 'curvaceous' shape of the sinusoid. Merriam-Webster (http://www.m-w.com) supports me in this. So what's wrong with the well-known, logical and sensible explanation?
—Herbee 20:56, 2004 Mar 25 (UTC)

It's not that Webster is wrong, per se—the English "sine" does come from sinus—but the reason why sinus was used is apparently much more interesting than you assume. My source is Carl B. Boyer, A History of Mathematics, 2nd ed. (see references). He writes (p. 209):

...Thus was born, apparently in India, the predecessor of the modern trigonometric function known as the sine of an angle; and the introduction of the sine function represents the chief contribution of the Siddhantas to the history of mathematics. Although it is generally assumed that the change from the whole chord to the half chord took place in India, it has been suggested by Paul Tannery, the leading historian of science at the turn of the century, that this transformation of trigonometry may have occurred at Alexandria during the post-Ptolemaic period. Whether or not this suggestion has merit, there is no doubt that it was through the Hindus, and not the Greeks, that our use of the half chord has been derived; and our word "sine," through misadventure in translation (see below), has descended from the Hindu name, jiva.

The "(see below)" I think refers to a much later section (p. 252) on translations of Arabic mathematics in Europe in the 12th century. There, Boyer writes:

It was Robert of Chester's translation from the Arabic that resulted in our word "sine." The Hindus had given the name jiva to the half-chord in trigonometry, and the Arabs had taken this over as jiba. In the Arabic language there is also the word jaib meaning "bay" or "inlet." When Robert of Chester came to translate the technical word jiba, he seems to have confused this with the word jaib (perhaps because vowels were omitted); hence, he used the word sinus, the Latin word for "bay" or "inlet." Sometimes the more specific phrase sinus rectus, or "vertical sine," was used; hence, the phrase sinus versus, or our "versed sine," was applied to the "sagitta," or the "sine turned on its side."

By the way, assuming an etymology of sinus for sine because of the "curvaceous shape" of the sine (from the other meaning of sinus for "curve," in particular the curved shape of a draped toga or garment) is probably an anachronism. Plots of the sine function ala analytic geometry didn't come until centuries after Chester. On the other hand, Chester may have mistakenly thought that "bay" alluded to the subtended arc; I'm just speculating, though. Steven G. Johnson 22:18, 25 Mar 2004 (UTC)

A little note in arabic. the letter representing V in arabic is very rarely used. The reason for this is i think its not actually ORIGINALLY recognized. Not even in the alphabetic of the language. I think it was the simplest thing to translate the letter "V" into a "B". further more jiba is hard to pronounce in a sentince describing an angle, and therefor might have led the arabs changing the order to better siute their pronounciation. Also the creation of new vocabulary of the word "bay". Also taking into account all of the other trigmetical words are synchronized in a way. Its all speculation but the following example in pronounciation should clerify things:

short forms used when talking math, like tan : tangent
sin : jaib : ja
cos : jata : jata

(recently the extra arabic letters have been un-officialy imported into english letters. using this i can represent the three variations of the english letter T into T , 6 , '6(the " ' " representing 6 but with a dot) as arabicly pronounced letters) based on this

tan : '6il : '6a
cot : '6ata : '6ata

I hope the resemblense can be noticed. this is also implemented in the last 2 of the original 6 common trignometical functions. Another example of missing arabic letters other than "V" is the letter "P". Which you can sence in 80% of the english speaking arabs, when talking to them you can hear words like "broblem" and so forth.

Note that the "versed sine" is 1–cos(&theta) = distance from the center of the chord to the center of the arc. I'm guessing that rectus and versus here refer to what we would now call the y and x coordinates, assuming that they originally drew a circle and measured the angle from the horizontal...Boyer doesn't say, however. Further evidence for this is the fact, according to the OED, that "sagitta", originally a synonym for the versed sine, is also an obsolete synonym for abscissa. sagitta is Latin for "arrow", and according to the OED's citations this is a visual metaphor for the versed sine (if you see the arc as the bow, the chord as the string, and the versed sine as the arrow shaft). Note that Wikipedia could use a short entry on versed sine. Steven G. Johnson 21:55, 25 Mar 2004 (UTC)

If you search for "jaib sinus" online, you find a number of other sources that confirm Boyer's etymology, notably:

• Eli Maor, Trigonometric Delights (http://www.pupress.princeton.edu/books/maor/), ch. 3: "Six Functions Come of Age (http://www.pupress.princeton.edu/books/maor/chapter_3.pdf)" (Princeton Univ. Press, 1998).
• Trigonometric functions (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html) (MacTutor History of Mathematics Archive)
• Amartya Sen, "Not Frog, But Falcon (http://www1.timesofindia.indiatimes.com/articleshow/33808779.cms)", The Times of India (Jan. 9, 2003).
• Prof. L. A. Smoller, The birth of trigonometry (http://www.ualr.edu/~lasmoller/trig.html)

Maor attributes the sinus translation to Gherardo of Cremona (c. 1150) instead of Robert of Chester (although he doesn't explicitly say Gherardo was "first"). Boyer, however, describes how both Robert of Chester and Gherardo of Cremona, along with several others, were contemporaries who were gathered together in Toledo by the archbishop there, where a school of translation was developed. Boyer says that Robert made the first translation of e.g. the Koran and of al-Khwarizmi's Algebra, among other things. Boyer also says, however, that most of these works are not dated, so it is possible that there is some uncertainty over who first translated the trigonometric work.

Maor also says that, although the first use of half-chords was in the Siddhanta, the first explicit reference to the sine function was in the Aryabhatiya a century later. There, Aryabhata the elder uses the term ardha-jya, which means "half-chord", which he later shortens to jya or jiva.

Some of these online works, especially the Maor book, seem quite nice. It would be great if some of this information could make its way into Wikipedia. —Steven G. Johnson 02:48, Mar 26, 2004 (UTC)

About CSC (http://en.wikipedia.org/wiki/ComputerSciencesCorp)

Is there anyone from Computer Sciences Corporation? (http://www.csc.com)

perhaps you will be willing to write an article that introduces your company :)

"Using only geometry and properties of limits, it can be shown that the derivative of sine is cosine and the derivative of cosine is negative sine. One can then use the theory of Taylor series to show that the following identities hold for all real numbers x:"

This statement is false. I show a proof for this that does not use geometry or properties of limits on the trig identity article. I am removing the word only. --Dissipate 06:11, 28 Jun 2004 (UTC)

I assume you're talking about the "linear differential equations" approach to prove d(sin x)/dx = cos(x). I have some comments about that, but first, I would point out that I think you misread what I wrote. If you read it carefully, all that it claims is that "There exists a method which shows that the derivative of sine is cosine and of cosine is negative sine, and which only uses geometry and the properties of limits". I made no such claim that this method was ITSELF the only method to solve the problem. I only made an existence statement, not a uniqueness statement.

But it doesn't matter much, because even the proof you suggest uses geometry and limits. Moreover, ANY PROOF MUST USE EACH OF THESE, for the simple reasons (1) if sine and cosine are to be defined indepedently of infinite series, or analytic methods, say, then they have to be defined geometrically; in my method, they are the real and imaginary parts of a point on the unit circle (or, x- and y-coordinates) parametrised by the circle's arc length, (2) the problem asks us to find a derivative...since a derivative is defined using limits, by definition we must use limits at some point.

I don't think your method at the other article is wrong...I think it's been misinterpreted. The point at which you use geometry and limits in one fell swoop is when you sneak in the result on the solutions of linear diff eqs. The problem here is that to prove (check) that this is the right solution requires knowing the derivatives of sine and cosine, so we're assuming what we're trying to prove. But, the argument is important and instructive. The diff eq itself along with the initial conditions can be "proven" informally using physics/vector ideas, (see Tristan Needham's book), i.e. the eqs come from a geometric conception of sine and cosine independent of analysis. Then, roughly the same argument (it's probably a bit different) will get you d(sin x)/dx = cos(x), using only properties of limits, or at worst, elementary properties of derivatives. Then, you have another "definition" of sine/cosine -- you define them as the solns of the IVP, and this definition is justified by the informal physics/vector analogy. It's an important way to look at it.

Revolver 09:26, 28 Jun 2004 (UTC)

Revolver: you are right, I misinterpreted. I thought you meant only infinite series and those two limits specifically on the trig identity page.--Dissipate 03:02, 29 Jun 2004 (UTC)

Multiple of 3 deg

Is it true that you can calculate the exact value of the sin or cos of any multiple of 3 deg (π / 60), as stated in this article? This looks to me like a typo for multiples of thirty degrees, which I would agree can be done by hand. Can anyone work out sin (39 deg) exactly by hand? (no calculators allowed) Ian Cairns 23:58, 4 Jul 2004 (UTC)

This is done at exact trigonometric constants, although I haven't personally checked every identity. Revolver 04:21, 7 Jul 2004 (UTC)

Image problem?

In Internet Explorer 5.50 for Windows 95, the "all six trig functions" image thumbnail appears black with only the colored lines and trig function names visible (not the circle or black letters). Could this be a transparency problem? Weird thing is, the full-size version looks fine. - dcljr 05:46, 11 Oct 2004 (UTC)

Other problems

In the Computing section, I've corrected the mistaken claim that calculators use "the Taylor series described below or a similar method" to calculate the trig functions. Actually, the method they use is nothing like a Taylor series (as far as I know); it's called the CORDIC method.

I've never looked at this in detail, but if I understand correctly, CORDIC is mainly used in only very low-end embedded hardware and FPGAs that lacks multiplication units.

Compared to other approaches, CORDIC is a clear winner when a hardware multiplier is unavailable (e.g. in a microcontroller) or when you want to save the gates required to implement one (e.g. in an FPGA). On the other hand, when a hardware multiplier is available (e.g. in a DSP microprocessor), table-lookup methods and good old-fashioned power series are generally faster than CORDIC.[1] (http://www.dspguru.com/info/faqs/cordic.htm)

So, I don't think your edit is correct.—Steven G. Johnson 23:40, Oct 27, 2004 (UTC)

I found a review paper (Kantabutra, 1996) that gives an overview of the different techniques, with many references. On modern general-purpose CPUs, a combination of coarse table lookup with some kind of polynomial approximation or interpolation seems to be the dominant technique. I've updated the article accordingly. —Steven G. Johnson 01:14, Oct 28, 2004 (UTC)

More importantly, consider this excerpt from the same section:

Using the [[Pythagorean theorem]],
''c'' = &radic;(a<sup>2</sup> + b<sup>2</sup>) =
&radic;2. This is illustrated in the following figure:
<br>

Therefore,
:<math>\sin \left(45^\circ\right) = ...

Umm... where's the figure? (The article's been this way since at least January 2004! Am I missing something here?) - dcljr 06:27, 11 Oct 2004 (UTC)

Arabic words Jiba and Jaib (and TeX)

I have added the Arabic spellings of Jiba and Jaib (they're the same). Thanks for fixing my link to Arabic alphabet I will now slowly help with TeX issues.

Query

Sinusoidal redirects here for some reason, although it's a hearing problem. Is this a bad redirect, or does Sinusoidal exist as a mathematical term as well?

Sinusoidal is indeed a mathematical term (it's even non-mathematical English: repeatedly wavy or curvy). It means roughly: varying in the manner of a sine wave.

The article at tinnitus does intend to point here (via redirect at sinusoidal) but what they are trying to say isn't clear to me. It could mean that the sound is a pure tone (pure tones have intensity verse time that is a sine function) but it is saying something about beats... I'm not sure what they are trying to say. Someone who knows what they are trying to say may want to clear that up.RJFJR 00:47, Dec 30, 2004 (UTC)

"Generality"

The article says that

They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of coordinates of points on the unit circle, or, more generally still, as infinite series, or equally generally, as solutions of certain differential equations.

This had me scratching my head wondering in what sense the later definitions were more general. I'm guessing this is referring to the domains of the functions: 0 to pi/2 in the first case, all reals in the second, and all complex numbers in the third and fourth. This was not clear though. Josh Cherry 02:36, 15 Apr 2005 (UTC)