User talk:Keith Davie

Reference desk

Please do not troll the reference desk, or use it to self-promote. --Kinu ^t/_c 02:46, 12 May 2008 (UTC) I feel it is worth retracting my comment. Perhaps I was somewhat suspicious of your motives, and I apologize. --Kinu ^t/_c 00:10, 13 May 2008 (UTC) I am thick skinned and really did not take offense. And I make no claims to a personal diplomacy. I full well anticipated initial resistance to my model and the two functions and expect more. After all, I resisted the model I encountered nearly a half century ago.. However, after years of playing in the sandbox for my own amusement, I have organized my observations and have presented an alternative to that model and not just a complaint about it. I believe that is what one is supposed to do in mathematics and the sciences.

Kindest regards, Keith Davie (talk) 04:58, 14 May 2008 (UTC)Keith Davie

==Re: Functions RD discussion May 12 2008== Maelin, thank you for your responses. I am not missing the point.

I have not seen it established that mathematics is complete, that all its tools have been discovered and accepted and that no more exist or can exist, nor shall further uses shall be discovered. I am not ready to suspend reason and take what you say on faith.

In addition to three instances which I cited in the referenced discussion in which there is no evidence in mathematics of Davie's functions or his Trigonometric Triangle model from another source, I will add that I was not taught the functions of the Trigonometric Triangle model which I have presented as a solution to the images offered at Wiki – two triangles, not one, or a first quadrant angle/triangle which extends into the fourth quadrant. Nor have you or anyone else provided any indication that mathematicians have found these tools useless in the past. There is no mention of the functions; where is the reference for your claim?

As to usefulness, if I were to present you with two functions recognized by math and which share a point (Z in my model), the exsec and tan expressed linearly/geometrically as line segments, what could you construct using a straightedge and compass?

If I were presented with the two functions of my model which share point Z, i.e. the dav and the tan expressed linearly/geometrically as line segments, using a straightedge and compass I could construct the complementary angle, the given angle, the Trigonometric Triangle, the interior rectangle representative of the rectangle of the given angle AND the unit circle. In other words, I could construct the whole enchilada. So given that usefulness, if you cannot do the same with the functions of the preceding paragraph, then perhaps we are discussing the wrong function's usefulness.

I agree that davie's functions are a) easily constructible from the other functions, but as I earlier stated at the referenced thread, the tan, cot, sec and csc also are easily constructible from the sin and cos. Are you willing to dismiss the tan, cot, sec and csc on the same grounds that you dismiss Davie's functions? The argument must be applied across the board. Like the four mentioned functions, Davie's functions are described in the theoretical terms of sin and cos as sin^2/cos and cos^2/sin. See Not Quite Nine Functions for Davie's definitions; also, Function Cotillion linearly/geometrically demonstrates that exponential functionality.

The only arguments I have seen offered here at Wiki have been non-arguments, false arguments and a straw man or two.

That it has been suggested that Davie's functions and model would add a layer of complexity to teaching is not an argument, it is an excuse of apparently lazy teachers.

Please consider looking at the videos at Google of my Trigonometric Triangle Trilogy again. Yes, I no doubt have made a minor error or two, but they will be minor and correctable within the model and accepted thought. And please watch the Function Cotillion. Consider what is presented in its entirety. Especially consider the last dance, Strut And Exponential Leap. There is more in these vidoes about which one can be positive than about which one can be negative.

I was not introduced to these tools by mathematics. I stubbed my toe on them while exploring something else in geometry. I now have studied them for over three decades. I am intimate with them.

Is it your argument that these tools should be something which needs be discovered again and again by students because the gods of mathematics, the trig teachers, are either unaware of or in denial about them?

I am especially interested in what you can geometrically construct when given the line segments tan and exsec.

Kindest regards, Keith Davie (talk) 00:00, 20 May 2008 (UTC)Keith Davie

Hi Keith. Hope you don't mind if we continue this discussion here on your talk page, I've spoken to some people who insisted on posting cross-replies on each other's talks which just gave me a headache. I'll try to respond to each of your points in turn.

• It is true that mathematics is not complete, and that new theorems are proven all the time. But this is not quite the same as saying that new functions are waiting to be discovered. New functions are dead easy to create: simply write a definition. Here's one: $\textstyle f:\mathbb {R} \rightarrow \mathbb {R} ,f(x)=x^{7}+3x^{4}-{\frac {1}{x^{2}+e^{x}+2}}$. Mathematics is not so much about discovering new functions as about discovering properties of functions and the spaces they operate in. Anyone can make up a function, the interesting part of mathematics comes from investigating how those functions work and what properties they have.
• I'm afraid I don't understand what you're asking in the paragraph starting with "In addition to three instances", could you please rephrase it?
• I agree that you given various beginning states, you could construct the triangle et al with compass and straightedge. Compass and straightedge constructions are interesting as a topic in themselves, but this does not mean that everything constructible with compass and straightedge is generally useful.
• Certainly, tan, cotan, sec and cosec are easily constructible from the functions sin and cos. But that wasn't my point. My point was that your "davie" functions are not ONLY easily constructible, but ALSO not particularly useful in general mathematical applications. If you were writing a paper in which they were frequently used, it would be perfectly acceptable to name them, define them and use them. But they are not of widespread usefulness to mathematicians, which is why they have not been given universal names.
• I'm afraid I can't watch the videos at the moment, but I will try to give them a look if I get time later. I hope you don't interpret my arguments as claims that your investigations or functions are worthless. I'm merely trying to explain that just because a function doesn't have a universal name, and isn't taught to school children, doesn't mean that mathematicians have never heard of it or know nothing about it. We never teach schoolchildren about the function $\textstyle g:\mathbb {R} \rightarrow \mathbb {R} ,g(x)=x+73052$, but this doesn't mean that it is a new function with new properties and that mathematicians have been "denying its existence". It is simply one of infinitely many easily constructible, largely uninteresting functions on real numbers. Certainly, your functions are more interesting that my g. But still, I hope you can see my point.
• Your continued characterisations of mathematicians and teachers as some sort of sinister cabal, intent on suppressing the truth of the davie functions and poisoning young minds with their lies, are counterproductive and disincline others to listen to your actual mathematical arguments. Please stop pretending that the mathematical world is denying the existence of these functions, and accept that they are not taught to children simply for pragmatical reasons.

-Maelin (Talk | Contribs) 04:11, 20 May 2008 (UTC)

I'm afraid I don't understand what you're asking in the paragraph starting with "In addition to three instances", could you please rephrase it?

When I say that mathematics has not recognized the functions it is because what I have found is the absence of these functions, as I cited with three instances where mathematics obviously does not recognize these functions. I realize absence is not proof. The only other thing I have found is unsupported claims that mathematics has considered the functions and has dismissed them. These observations (of absence and lack of supporting evidence for the claims made) can be taken as fact, can be refuted with supporting evidence or can be made into a conspiracy theory. I view my statements in that matter as statements of fact awaiting refutation. I have had responses which describe the properties of the functions in useless ways and not as described by Davie.

Certainly, tan, cotan, sec and cosec are easily constructible from the functions sin and cos. But that wasn't my point.

Yes, but the point is that constructibility must be applied equally. Usefulness is in the eye of the beholder and must be applied equally and not out of ignorance. Yet the exsec, a function I know to be recognized by mathematics, when combined with the tan, will construct every angle including the 'given angle', but without discriminating for the 'given angle'. Mathematics recognizes that easily constructible and useless function. So why the exception for the exsec, except that its uselessness does not agree with the argument against Davie's functions, which have at least one recognizable use.

Mathematics is not so much about discovering new functions as about discovering properties of functions and the spaces they operate in.

Yes. Unfortunately, you can't watch the videos at the moment, so it follows there is really not much to discuss with you at this time regarding the properties of Davie's functions or the spaces they operate in as covered in those videos. Understand that I have discussed the properties (theoretic, geometric and numeric) and spaces of Davie's functions in my videos and am prepared to discuss them further.

FYI, the three videos in the trilogy run between 15 to 18 minutes each. The Function Cotillion, which devotes itself to the spaces in which the functions operate, runs in less than three minutes.

Thanks for your response. Given the exceptions which must be made, I would have preferred to have seen some supporting evidence for your uselessness claims. Since I have found and described a use for Davie's functions, which is similar to a use of other functions so situate, yet is uselessness with the mathematically recognized exsec in that situation, I am not ready to take the 'uselessness' argument on faith.

~~Keith Davie

You still seem to think that mathematics doesn't recognise the functions. There is a very big difference between "recognising a function" and giving it a universal name. There are only finitely many names to go round, but infinitely many functions to think about. So mathematicians only give universal names to the functions that are proven to be significantly and widely useful or interesting to many different mathematicians in many fields. After four years of studying pure mathematics, I've never seen a mathematician, a mathematical paper, or a mathematics textbook talk about my function g that I defined above. But still, I wouldn't claim that "mathematics has failed to recognise it," and then name it after myself and demand it be studied in schools.

For that matter, mathematics recognises your functions by default. Mathematicians will immediately recognise any function that is well-defined and say, "yes, that is a function." The next thing they will say is, "what about it?"

"Yet the exsec, a function I know to be recognized by mathematics, when combined with the tan, will construct every angle including the 'given angle', but without discriminating for the 'given angle'."

I don't follow. In trigonometric applications, the exsecant function doesn't construct angles, it constructs a length given an angle. Note that the exsecant article points out that since calculators have removed the need for precalculated trig tables, nobody really bothers with the exsec function any more. Note that school children are not explicitly taught the exsec function.

If you want to investigate the properties of the functions you have described, you are welcome to. Nobody is disputing that. What is disputed is a) your claim that "mathematics has failed to recognise" the functions, which is patently untrue, and b) your expectation that simply because you have named these functions (after yourself, no less), they should be taught to school children.

-Maelin (Talk | Contribs) 05:42, 21 May 2008 (UTC)

Maelin, I understand your argument. It is a distraction from Davie's Trigonometric Triangle model and Davie's functions. You don't have the time to watch the videos. Fine. The videos cover the properties of the functions and the spaces they operate in in terms of theory, geometry and trigonometry, which would be valid concerns if not covered. You raised those concerns. They are covered. I reviewed all four of my videos this morning before coming here to make sure those concerns were covered in my videos. The eight functions of Davie's Trigonometric Triangle are discussed in terms of theory, geometry and trigonometry, not just Davie's functions.

That you 'don't follow' is understandable as you have mischaracterized what I said above. That amounts to a straw man argument. The situation presented to you was the geometric (linear as line segments) interaction of two functions, exsec and tan. The situation was not about just the exsec or just the tan, but about their geometric interaction and usefulness compared with the geometric interaction and usefulness of the tan and dav. The exsec and tan cannot discriminate through their geometric interaction a construction of the given angle. The tan and dav do interact geometrically to allow the construction of the given angle.

And you are right in that the exsec was not taught when I was a student a half century ago. But the exsec was recognized by math through reference and definition, unlike Davie's functions by any name. So don't get hung up on the false argument about recognition. Consider instead Davie's Trigonometric Triangle model, Davie's functions and the six recognized functions in relation to that model. And consider that model in relation to the models which math uses and teaches, as evidenced by Wiki's accompanying illustrations at Functions and Trigonometry. From those choices at Wiki it is apparent that mathematics does not have a unified, elegant, single, first quadrant (acute angle) Trigonometric Triangle which it recognizes either, much less one which has application through the four quadrants when given the usual considerations. So let us not engage in false arguments. And let us also limit our discussions of functions to the functions of geometry/trigonometry, the functions against which Davie's functions must be compared, and not the functions of other disciplines.

That I have named the model and the functions after myself is academic privilege which I claim based in my scholarship regarding those functions, and the naming occurs finally in the third video to overcome the cumbersome references unnamed function and unnamed co-function. Math didn't give me a name to use so I picked one. Apparently I got there first. Get over it.

That Davie's functions are not formally recognized by mathematics by any name is a fact. Thus, anyone searching mathematics for those two functions by any name or even no name will needs reinvent them every time for mathematics, unlike with the exsec and vers, makes no reference to those two functions and offers no definition of those two functions. That the exsec and vers are formally recognized by mathematics is a fact, even if either or both subsequently have been deprecated. To argue about those facts is a distraction from that which you don't have or won't make time for. You could have watched one of the videos in the time it took you to present the distraction and straw man argument.

In your response you have avoided trying to explain the exceptions upon which your argument must rely. In spite of your finding time for the above response, credit that lack of explanation of the exceptions upon which you argument must rely on your lack of time. The scholarship of my videos a priori cover the concerns which you raised about the properties of the eight functions and the spaces in which they operate.

Kindest regards, Keith Davie (talk) 17:45, 21 May 2008 (UTC)Keith Davie

It appears that I have missed your point, then. I thought your contention was that mathematics "does not recognise" these functions (a meaningless phrase; mathematicics is a study of functions and spaces, any function that is well defined is automatically recognised, in any meaningful sense of the word) and that only by naming them are they recognised. I thought you wanted us to use your triangle construction as a more concise demonstration of the trig functions than the diagrams given in the trigonometric functions article. If this isn't the case, then I'm sorry, but that is the impression that I derived from your previous posts.

You continue to posit the existence of some entity known as "mathematics" that has the power to recognise or not recognise functions, and this is a fantasy borne of misdefinition. Mathematics is a study, a field of inquiry, an intellectual pursuit. It is not a thinking entity capable of recognising concepts. It is not a collective noun that encompasses mathematicians, mathematical textbooks, and mathematical papers. If you mean to say that mathematicians do not "recognise" your functions and do not teach them to students, say so. If you mean that your functions are not defined in literature, say so. Then, once you have stated exactly what your issue is, argue why things should be different. These "straw man" arguments are the result of your contention being unclear in this matter.

I have now just watched your "Function Cotillion", and whilst I found the dance metaphors charming, I feel no more enlightened as to the point you are trying to make. The behaviour of all the functions on the triangle as the angle varies are not mysterious, they are perfectly well-understood and could be graphed by a first year undergraduate. If there are more videos you want me to watch, please link them in here and I will watch them tomorrow night.

-Maelin (Talk | Contribs) 09:48, 22 May 2008 (UTC)

First, regarding 'mathematics', yes, a field of study is one experssion. I prefer 'discipline'. Within that discipline there are texts of various kinds. In referencing the texts of the discipline Davie's functions, by any name, are not formally recognized.

The Illustration in Wiki at Trigonometry and Exsecant is of a given angle in the first quadrant with an inelegant extension to the fourth quadrant (segments OB and CB which form a right triangle OCB). What is the need for that fourth quadrant extension to describe the given angle's functions?

Interestingly, while exsec has been deprecated because of the calculator, it is still formally recognized in the discipline of mathematics through references such as the one here being discussed, as found in the literature of the discipline.

Again, at Versine in Wiki the given angle is first quadrant acute and right triangle OCB of the illustration provided by Wiki extends into the fourth quadrant with an angle which is greater than 270˚.

Since you say The behaviour of all the functions on the triangle as the angle varies are not mysterious, they are perfectly well-understood, and since I have never found literature within the discipline which describes Davie's functions by any name and therefore which brings into question the claim that the function is perfectly well-understood, please be good enough to describe the understanding of the behavior of the function ZW in Davie's functions (function EC in the illustration which Wiki provides at Trigonometry and Exsecant) and please provide a reference to that understanding within the literature of the discipline. Clearly such a discussion will include the similarities and dissimilarities when compared with the behavior of the other functions of the given angle.

Thank you.

Kindest regards, Keith Davie (talk) 15:27, 22 May 2008 (UTC)Keith Davie

And there lies the crux of it. You don't see that mathematics can study a function unless it names it. Would you claim that my function, $\textstyle g:\mathbb {R} \rightarrow \mathbb {R} ,g(x)=x+73052$ is not recognised? I've never seen it in mathematical literature, never been taught it in 13 years of maths at school or four years of maths at university. Have I discovered a hitherto unknown function? Have mathematicians failed to recognise it? Do they have a less-than-complete understanding of it? Should more time and resources be devoted to it? Should it be taught in schools? Do we need a Wikipedia article about it? If, as any reasonable person would, you answer "no" to these questions, please explain why my function, g, is different from your functions in this respect. What makes your functions worthy of a universal name, and worthy of study in maths classes and mathematical literature, while my g is not? This is a short response as I haven't much time at the moment, but I will hopefully get a chance to respond in more length later. Maelin (Talk | Contribs) 20:41, 22 May 2008 (UTC)

...let us also limit our discussions of functions to the functions of geometry/trigonometry, the functions against which Davie's functions must be compared, and not the functions of other disciplines.

Consistency in the application of rules is paramount to the success of disciplines.

Mea culpa, I realized after posting that you have already answered my query.

Your claims that mathematics has "failed to recognise" the functions yielding the lengths of WZ, VX and XY are somewhat unfounded. They are easy to define, as given the angle θ, we can prove that WZ = OZ - OW = sec(θ) - cos(θ), that VX = 0X - OV = cosec(θ) - sin(θ) ...

Which is one definition. And yes they are easy to define. And no, that definition is not found in the literature of the discipline. I agree that ease of definition is not coincident with recognition.

So too the easy definition can be made, given the angle θ, ZW = vers + exsec and XV = covers + excsc.

While both define those segments, neither is the best definition since neither reflects the manner in which the functions act in those spaces. Ease of definition is not necessarily a goal.

Consider the sin is derived from the side of the right triangle which is opposite the given angle being divided by the hypotenuse of the right triangle, and the cos derived from the side adjacent the given angle divided by the hypotenuse. They are quotients.

Consider the tan and cot are derived from the adjacent sides of the right triangle be devided into each other. They are quotients but are not a part of that triangle from which they are derived except at point Y.

Consider the sec and csc result from the hypotenuse being divided by the cos and sin. They are quotients and they extend beyond the triangle of the sin and cos except when they are at their least.

Each of these functions are quotients derived from one right triangle.

Now consider, this discussion of Davie's functions in Not Quite Nine Functions. (Google Video search term: Trig > Not Quite Nine Functionss)

While determining WZ could be viewed as a simple subtraction (sec - cos) we are going to maintain our triangular approach. We observe that WZ, like the cosine OW, appears on the line of secancy. Also WZ is one of the adjacent sides of triangle ZWY, YZ is the hypotenuse of that triangle and YW also is an adjacent side. Since segment YW is a side of two triangles (ZWY and OWY), one of which triangles WZ shares with segment YW and one of which WZ does not share with segment YW, and as the earlier quotients derived from divisions of the same triangle, not two, multiplication is here indicated. The lines to be multiplied are YW times YZ, (sin Angle ZOY times tan Angle ZOY). Interestingly we know that the tan of an angle is itself the quotient of the sine divided by the cosine. Thus when multiplying sine times tangent we get sin^2/cos, which is to say we get a quotient, which we identify as line segment WZ. Making similar observations about the other unnamed co-function (VX) as made about segment WZ, VX can be seen as the product of VY times YX or the product of the transposed cosine of Angle ZOY times the cotan of Angle ZOY, which can be expressed also as the quotient of cos^2/sin, which we identify as line segment VX.

(Note: for consistency since I wrote that, I have since tried to state line segments in the order such that the first letter resides at an acute angle and the second letter resides at the right angle of the triangle on which the line segment resides.)

One point here being I have maintained the consistency of determining the functions from the sides of right triangles and that ZW and XV, like the other functions, are quotients, not sums or differences.

As the sin acts as a side in triangles OWY and ZWY it should not be surprising that we find sin^2 in the quotient.

And the quotients are expressed such that they demonstrate the exponential nature of the function ZW. That exponential nature is demonstrated further by the charming metaphors in the video, Function Cotillion. In that video we observe first that points Z and W are in coincidence with Y on the circumference of the unit circle at 0˚, which is the first angle of the trig tables and that coincidence is supported by the trig tables and demonstrated geometrically. We may next observe as the angle in the video increases from 0˚, point Y follows the arc of the unit circle and points Z and W move away from each other and from the fixed point of the circumference.

But they move away from that fixed point at different rates as the angle increases through the first quadrant to 90˚. Point W travels one unit from its starting point to point O during that increase. The rest of the function to point Z goes from one unit away from point O at its start to an undeterminable distance away from its starting point, an exponentially greater distance during that increase. While the points of the line segments of all the functions move away from and towards each other as the angles increase or decrease, none of the others move in the way that the exponentially derived quotients moves. The movement of the points Z and W as well as points X and V over a range of angles demonstrates the exponential nature of the function, that it is more than the simple subtraction or addition of line segments.

So the functions to which I refer as dav and codav are exponential when derived from the right triangles as a quotient, like the other six major functions are derived from right triangles as quotients, and not as simple subtractions or additions.

We might say I have been charmingly consistent in my definition of Davie's functions.

Kindest regards Keith Davie (talk) 01:11, 23 May 2008 (UTC)Keith Davie

The sum and quotient definitions are equivalent and thus there is no need to distinguish between them. You still refuse to answer my question, refuse to tell me why your functions deserve names but my g doesn't, and your distinction is arbitrary. Both your davie functions and my g are functions from the real numbers to the real numbers, yours merely have a more intuitive geometric definition. But, that's fine. Here is my new function, now trigonometric, f, defined for an angle θ in a triangle: $f:\mathbb {R} \rightarrow \mathbb {R} ,f(\theta )=\sin(\theta )+73052$. I can even outline a compass-and-straightede construction if you really want, not that it makes any difference how you define your function as long as all prospective definitions are equivalent.

Assume we are given a line through the origin, meeting at angle θ to the positive x-axis in the Euclidean plane. Draw a point of intersection, A of this line with the unit circle. Draw a line parallel to the y-axis passing through A. Measure 73052 unit lengths up the line and label this point B, label the point of intersection with the x-axis as C. f(θ) is defined as the length AC.

Now, explain to me why your functions deserve a universal name but my function f doesn't. Maelin (Talk | Contribs) 06:00, 25 May 2008 (UTC)

Maelin, if I were a mathematician I might be able to speak your function. I am not a mathematician. You have refused to answer my questions on why your definitions require their exceptions.

After over three decades I have become intimate with my unified, singular (universal) Trigonometric Triangle model, its functions and the functions commonly used to measure the given angle. I can speak to that.

While it may be that The sum and quotient definitions are equivalent in their numeric value's end result, it is how that value is determined which gives one insight into the manner in which the function occupies the space, i.e., how the tools operate and are best used. It can be demonstrated that a screwdriver can be used to the same effect as a bottle opener. However, that is not the best use of the tool and could damage the tool for its best use. Similarly, the sums/differences approach to the two functions in question is not the best use of the tools which those functions represent. Also, the sum/difference definitions do not adhere to the derivation of the definitions of the other functions as quotients of sin, cos, including the sin and cos themselves, nor are sums/differences determined in the same manner in relation to the right triangles as the quotients of the other functions, whereas sin^2/cos, cos^2/sin does meet that method of definition derivation. Further, the two functions in question fit the geometry of a unified, singular (universal) Trigonometric Triangle, a triangle which measures the functions of the given angle, a triangle which mathematics appears from its literature to be wanting. Hence the definition of the two functions in question as quotients demonstrates a consistency of application of the rules with which the other functions precede and confirm the model of definition derivation, which do not require exceptions to their model. Regards, Keith Davie (talk) 15:10, 25 May 2008 (UTC)Keith Davie

You have refused to answer my questions on why your definitions require their exceptions.

What exceptions? What definitions require exceptions? What do they require exceptions to? Please answer this before going on so I can understand where you are coming from. Maelin (Talk | Contribs) 01:44, 26 May 2008 (UTC)

It appears that we shall continue to dance in circles and get a no further than we are now. The exceptions I queried regarded the exsec not being derived from the sin and cos or as a quotient as the sin, cos, cot, tan, sec, csc, dav and codav, or its lack of geometric usefulness, a usefulness which is demonstrable from the other functions, and the sum of cot + tan as a function which does not yield a unique result from the unique input of a given first quadrant (acute) angle.

That mathematics does not have a unified geometric model for the measurement of the given angle (trigonometry) is obvious in the different examples of geometric models published at Wiki and in various texts of the discipline.

Thank you for the dance.

Kindest regards, Keith Davie (talk) 14:55, 26 May 2008 (UTC)Keith Davie

Keith, the only reason we aren't getting anywhere is because you won't answer the questions I ask and you don't phrase your requests clearly enough for me to fulfill them. If the exsec function were introduced now, it would not be given a universal name either. The only reason it has one is because it's useful to surveyors and astronomers, and because before the advent of calculators, trig tables were used and so more functions had to be included on trig tables to improve precision for engineers. We can continue this discussion, rationally, like two mathematically-minded people, IF you will explain why the davie functions are mathematically significant enough to warrant a name but my trigonometric function f does not.

By the way, mathematicians don't define trigonometric functions using triangles nowadays, anyway. The standard definitions are derived from power series due to complex analysis. Maelin (Talk | Contribs) 03:00, 28 May 2008 (UTC)

Maelin, I'd rather not put the cart before the horse. Let us start with the word trigonometry which I find defined as the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. At the core of trigonometry is triangle / measure. If, as you say, mathematicians don't define trigonometric functions using triangles nowadays, then mathematicians must not use trigonometry and the discipline should be renamed to something reflective of 'power series'. Of course, angles and their respective and unique triangles somehow are geometric, so how that geometry is taken into account in 'power series' is something which mathematicians will needs explain. I certainly can't.

And yet, like so many other references to trigonometry and the functions of the angle, the link at

http://en.wikibooks.org/wiki/Trigonometry/Using_Fundamental_Identities

clearly uses a right triangle as a model in its discussion of fundamental trigometric [sic] identities ... derived from the Pythagorean Theorem.

And we can observe from that illustration that if the hypotenuse c = 1, then a = sin angle CAB and b = cos angle CAB. I observe that this triangle is the geometric model which I encountered nearly five decades ago when I took trigonometry in high school. We identify in that model the sin and cos of angle CAB. Now what is missing from that model? The other functions of the angle which are discussed theoretically!

This brings us to Davie's Trigonometric Triangle model in Not Quite Nine Functions – link:

http://video.google.com/videoplay?docid=6267568814783610621&q=not+quite+nine+functions&ei=bds-SKOHJqCErAOEyOjnAw

The video at Google runs for ~17 minutes. What is there about Davie's model which keeps it from being THE triangle which measures the functions through their geometric and theoretic identifications, i.e., the cart which is behind the horse?

Is it that the model first establishes from the given angle a right triangle with the sec, csc and the hypotenuse on which the tan and cotan reside? Is it that the sin and cos are established later in the geometric construction? Is it that the complimentary sin is transposed to be located with the complimentary angle? Is it that right triangles are used throughout the model? What is the objection?

Regards Keith Davie (talk) 17:18, 29 May 2008 (UTC)Keith Davie