Talk:Trigonometric functions/Archive 1

Archive 1

Archive 2

Archive 3

"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)

Could someone add, when it is first mentioned, an explanation as to what the set of zero's actually is? ---Crusty007 23:11, 18 March 2007 (UTC)

Done. Good idea! Cheers, Doctormatt 01:46, 19 March 2007 (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)

The confusion is over f^-1(x) which always means the inverse of f, versus f(x)^-1 which means the multiplicative inverse. The ambiguity is when sin^2(x) entered convention not as sin(sin(x)) but (sin(x))^2, probably due to even more confusion with sin(x^2) Obscurans 13:21, 28 April 2007 (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 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, ch. 3: "Six Functions Come of Age" (Princeton Univ. Press, 1998).
• Trigonometric functions (MacTutor History of Mathematics Archive)
• Amartya Sen, "Not Frog, But Falcon", The Times of India (Jan. 9, 2003).
• Prof. L. A. Smoller, The birth of trigonometry

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)

The page on the "Birth of Trigonometry" gives a good explanation. To quote:

From India the sine function was introduced to the Arab world in the 8th century, where the term jya was transliterated into jiba or jyb. Early Latin translations of Arabic mathematical treatises mistook jiba for the Arabic word jaib, which can mean the opening of a woman's garment at the neck. Accordingly, jaib was translated into the Latin sinus, which can mean "fold" (in a garment), "bosom," "bay," or even "curve." Hence our word "sine."

With the reference being originally from "The Crest of the Peacock: The Non-European Roots of Mathematics, new ed. (Princeton, NJ: Princeton University Press, 1991, 2000), p. 282." --AJ Mas 05:49, 14 November 2005 (UTC)

Etymology of inverse functions

Is there any interesting history of the names "arcsin," "arccos," et cetera, that could be included?

About CSC

Is there anyone from Computer Sciences Corporation?

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)

It is true, although such calculations may not be feasible. This may be done by seeing that for x = 36°, -cos(x) = cos(4x), using the double angle formulas and solving the resulting quartic with Ferrari's method. Then one may use the half angle formula to obtain the sine of 15°, which is equal to the cosine of 75°. Using the double angle formula for cosines will obtain the cosine of 72°, and the subtraction formula (along with Pythagorean identities) gives the cosine of 3°, from which any multiple may be obtained. Scythe33 18:05, 30 August 2005 (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]

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)

relations

I don't think it is useful to give the relations before giving the definitions. The cannot be used as definitions as is claimed unless you define at least one. --MarSch 18:23, 24 Jun 2005 (UTC)

Law of Sines Edit Requested

I request that the Ambiguous case for the law of sines be addressed in the article. Guardian of Light 6 July 2005 13:39 (UTC)

Arctan with two arguments

Is it worth mentioning that many computer languages have an additional inverse trigonometric function, called atan2 in C, used for polar coordinate calculations? Provided x and y are not both zero, it is defined by atan2(y,x)=t where t is the unique angle in [-π,π] such that x=cos t, y=sin t. This would be of practical value for computer science students seeking quick online help, but I don't want to mess with a featured article. --JahJah 10:09, 20 August 2005 (UTC)

That description is not quite right. The function is not restricted to cases where x^2 + y^2 = 1. In all other cases it is not true that x=cos t and y = sin t (x and y can even be bigger than 1 or smaller than -1). Josh Cherry 13:28, 20 August 2005 (UTC)

You're right. Do you think it is appropriate for inclusion? --JahJah 13:37, 20 August 2005 (UTC)

I think it should be included. salte 13:17, 4 December 2006 (UTC)

Another vote for inclusion here. Arctan with just one argument sucks. It's like, so pre-computer-age. --Oolong 14:55, 27 January 2007 (UTC)

Already discussed in Inverse trigonometric function. —Steven G. Johnson 21:03, 27 January 2007 (UTC)

Cotangent

I found the formula:

$\cot {x}=\sum _{n=-\infty }^{\infty }{\frac {1}{x+n\pi }}$

but don't know how to fit it in the article. Any suggestions?Scythe33 22:15, 24 August 2005 (UTC)

Maybe over at Trigonometric identity? --noösfractal 03:40, 25 August 2005 (UTC)

rational sine/cosine

If a sine or cosine of a rational angle (in degrees) is rational, then it has to be either -1, -0.5, 0, 0.5 or 1. All the other sines and cosines of rational angles are irrational. Is this fact known and is it written in this article?

Algorithm for Implementation

The following page explains an algorithm used to implement COSINE and another for SINE.

Sine and Cosine Function

Is there are good way to include the algorithm and would it be of interest to include it, rather than simply leaving it as an external reference?

That's the Taylor series expansion, which is already described in the article. (By itself, it's not suitable for serious use in computation, however. See the section on computation.) —Steven G. Johnson 17:07, 14 November 2005 (UTC)

complex angles

i have a math reference dictionary that says sin is "best defined as a complex function by the power series..." you know what i'm talking about. do that mean you can have complex inputs? i.e. angles? because that just sounds absurd, it's like saying negative distance in the context of negative energy, only negative energy is actually predicted actually i remembered that sinh(x)=-isin(ix). i can rule out the possibility of the book making a mistake now.

Complex angles exsists. yes it is fully possible to calculate the sine and cosine values of a complex angle. T.Stokke 19:39, 12 July 2007 (UTC)

Properties and applications new image

Hi, I was messing around with a little trig function plotting program I was writing in VB. I noticed an interesting shape that loops and meets itself in the center.

This is the image.

Do you think this would be a good addition to add below the "Functions based on sine and cosine can make appealing pictures." picture on the page? I could add axes and chance the colour scheme etc to make it more appealing.

It uses the equation:

$(x(\theta )\,,y(\theta ))=\sin(5\theta )\,,{\frac {\sin(8\theta )}{\cos(4\theta )}}$

It also follows an interesting rule relating to the coefficients and the number of loops it makes.

Do you think this would be a good addition to the page? -- Haddock420 16:53, 17 December 2005 (UTC)

Yes, I think it would be nice. In case you're curious, you can use the trigonometric identity sin(2x) = 2 sin(x) cos(x) to rewrite the equation as

$(x(\theta )\,,y(\theta ))=(\sin(5\theta )\,,2\sin(4\theta )).$

Figures of this form are known as Lissajous figures. I hope you're not too disappointed that your discovery is not new; it's happened to me a lot. -- Jitse Niesen (talk) 18:16, 17 December 2005 (UTC)

Slope definitions

How does the slope definitions section get away with not actually giving the slope definitions? Hehe. My algebra is a little rusty, so I have probably forked up the signs here:

Sin theta = rise/radius = rise/sqrt(rise^2+run^2) = rise/|rise| sqrt(1+(run/rise)^2)

So we see that

The slope definitions:

• Sin theta = + or - 1/sqrt(1+m^-2)
• Cos theta = + or - 1/sqrt(1+m^2)
• Tan theta = m

This is of course consistant with

Tan theta = Sin theta/Cos theta = sqrt(1+m^2)/sqrt(1+1/m^2) = sqrt(m^2*((1+m^-2)/(1+m-^2))) = sqrt(m^2)

Could some trigonomexperts verify/fix and add this stuff? --Intangir 21:17, 3 January 2006 (UTC)

Incorrect series formulae?

I found that the given series formula for tangent produces significant error for larger values of x, even at high N. Using Matlab, I computed 100 data points over $[0,0.9\pi /2]$ with the series and plotted the resulting error vs. the built-in function. I did this in a loop with N incrementing. The series converged (the error function stopped noticeably changing after N=15), but it was not even close to agreement.

Even if this was round-off error on the part of Matlab (seems unlikely for N<50), the formula still seems to be missing a factor of 1/6. For example, when N=1, the coefficient for the term is $2^{2}(2^{2}-1)U_{1}/2!=4\cdot 3\cdot 1/2=6\neq 1$, while the page implies a coefficient of "1" for the first term of the series. And, isn't there supposed to be a $(-1)^{N-1}$ term somewhere?

I am not an expert on this, but it seems that the "up/down" function is an approximation of the Bernoulli number which is found in other treatments. Can anyone explain these inconsistencies?

69.124.9.75 15:10, 21 January 2006 (UTC) Brandon

The up/down numbers are related to the even Bernoulli numbers, but they are not an approximation.

You were completely right that the series from the article were incorrect. Strange though, as I seem to recall checking them against MathWorld. Anyway, I now copied the series from MathWorld and checked that the first couple of terms agree, and I also checked it against Abramowitz and Stegun.

Thank you very much for bringing this to my attention. -- Jitse Niesen (talk) 16:51, 21 January 2006 (UTC)

Additional Functions

I have seen mention of the "versed cosine", "coversine", "versed cosecant", and "excosecant" on what appears to be a Russian website. Should these be included? (I think they would complete the symmetry within the image presenting eight other trigonometric functions.) (Also, I think pairing "excosecant" with "vercosine", or even "coexsecant" with "coversine", would be more symmetric than pairing "coversine" with "excosecant".) Zeroparallax 09:05, 5 February 2006 (UTC)

These functions are rare enough in modern use that only versine and exsecant need to be mentioned briefly here, and their variants can be described on the linked pages. (Indeed, see versine already.) And, as far as I can tell, "coversine" is far more common than "vercosine"; language is not always symmetric, I'm afraid. —Steven G. Johnson 19:09, 5 February 2006 (UTC)

Recent edits to "History" section

I notice that User:Jagged 85 has been making substantial edits to the "History" section, emphasizing the contributions of Indian mathematicians. I'm concerned, however, because he/she is providing zero references for the information added, such as supposed developments by the Babylonians and developments of the Taylor series in India 400 years before Euler.

The old version adhered pretty closely to the account by Boyer (see the references), and made no mention of e.g. Madhava that I can recall.

Please provide sources for substantial new historical claims.

—Steven G. Johnson 06:11, 28 February 2006 (UTC)

Citations

I think this article requires more inline citaions; escpecially becuase it is a featured article. Kilo-Lima ^{Vous pouvez parler} 21:09, 11 March 2006 (UTC)

suggestions

One of the things this article seems to be missing is a graph on a cartesian plane of the tangent function. Also, the fact that the cotangent of an angle is the same as the arctangent of that same angle has not been made clear.

I have created a graph of tan(x) at Image:Tangent.svg if someone wants to include it in the article.

Image sizes

I just reverted a part of this edit. The edit message was "let users pick their own thumb size in user preferences unless there is a specific reason not to". I do think there's a specific reason in this case—the text in the thumbs should be readable for a default user (i.e. a user who has the default thumbnail size). –Gustavb 03:59, 2 April 2006 (UTC)

Dead link

Trigonometric functions of angles 0° to 90° by degree in the links at the end of article is dead.

Thank you, now fixed. -- Jitse Niesen (talk) 06:29, 3 April 2006 (UTC)

The trig hand link is also bad. Instead of showing the pic it scolds and warns, evidently expecting an image upload. I did not look at the source. translator 20:10, 13 July 2007 (UTC)

Oops

I accidentally messed up a math expression halfway down the page, I'm not sure what I did and I'm not sure how to revert, can someone please fix my mistake? --Monguin61 23:39, 12 April 2006 (UTC)

I fixed it, nevermind, sorry --Monguin61 23:55, 12 April 2006 (UTC)

Confused about divergence statement

What am I misunderstanding in this statement near the end of Unit-circle Definitions?

From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90 degrees)

Why does one not say that the functions CONverge (toward infinity) as θ approaches π/2? Their proportional difference is nowhere greater than it is near 0 nor smaller than it is near π/2, no? This relationship (however it might properly be described) is instantly apparent in a unit-circle diagram that normalizes the tangent to vertical, as an earlier version of the diagram seems to have done some years back.

66.32.190.156 07:33, 5 June 2006 (UTC) AldenG

Confusion resolved. From the article on convergence, it emerges that when two functions approach equality at a common limit-value, they are said to converge -- unless the limit value is infinity or negative infinity, in which case they are said to diverge.

66.32.177.66 16:23, 19 June 2006 (UTC) AldenG