Template:Featured article is only for Wikipedia:Featured articles.
Accepting the point that "not monotonic" is not sufficient for requiring a restricted ___domain for an inverse, I think that the phrase "both continuous and not monotonic" is not the solution. Not all the functions are continuous. I have edited to replace the phrase simply with the word "periodic". This is (much) more than sufficient to require a restricted ___domain for the inverse, but has the benefit of consistency with other reasons for restricting the ___domain. --Richard Jones 20:54, 8 Mar 2004 (UTC)
This was just added, and I have to admit I don't really understand it:
- An alternative is to remember that sin starts at 0 and grows to 1, cos starts at 1 and shrinks to 0, and tan starts at 0 and grows to +∞. That avoids the requirement of remembering what the the adjacent, opposite and hypotenuse are called. For word-related mnemonics for remembering what the functions do, for sin one could imagine a holy spring in equilibrium (the total forces at 0), and one sins by punching the holy spring so it moves away, and the forces on it approach 1. For cos (pronouncing cos as caus) one could imagine a spring which starts off not moving, and causes a vibration, so the total forces on the spring start off as 1, going down to 0 as the spring relaxes. For tan, one could imagine when someone is having a sun tan, the photons start at the sun, and their distance from the sun practically increases to infinity on the way, because they travel so far.
This might work for the author, but does it make sense to anyone else? What's a "holy spring" anyway? moink 17:52, 14 Dec 2003 (UTC)
- A holy spring, or sacred spring, was just supposed to be a spring, which would be sinful to hit. I've shortened it a bit, without any puns on sin, cos or tan.
- Thanks Cyp, I like it better now. Mnemonics are often very personal things. moink 20:08, 14 Dec 2003 (UTC)
There used to be an entry here that the trigonometric functions for angles => 90° had not been dealt with. I am putting this reminder here. That surely comes before trig identitites. RoseParks
I removed it because the unit circle section takes care of all angles. The section and graphic could use refinement, but that's what the wikipedia is for :).---- I see box for a graphic of a unit circle, but no graphic? Anyone else see it?? RoseParks
I'm trying to send the gif, but the guy I'm supposed to email it to is having trouble recieving it. I'm going to refine the graphic and try again later.
which notation is more common for inverses: arcsin or sin^-1 ? Which came first? -- Tarquin 12:00 Mar 6, 2003 (UTC)
Id like somewhere to link to (for an article on the derivatives of trig functions) which explains some various algebraic rules of trig functions, such as those governing sinx(x + h). Maybe Ill have to do it myself. Pizza Puzzle
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)
About CSC
Is there anyone from Computer Sciences Corporation?
perhaps you will be willing to write an article that introduces your company :)