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Per 6/49, the chances of winning the jackpot is roughly 1 in 14 million. However, lotto 6/49 consists of matching 6 numbers from a box of 49 numbers. So therefore, shouldn't the odds of winning the jackpot be 1 in (69 x 68 x 67 x 66 x 65 x 64) or 1 in 86311779840? Where have I gone wrong? Acceptable ( talk) 00:50, 12 May 2008 (UTC)
Oh ok, haha I don't know why i used 69 and 68's instead of 49 and 48's. Thanks. Acceptable ( talk) 22:28, 12 May 2008 (UTC)
The article at http://en.wikipedia.org/wiki/Trigonometric_function states, "Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers."
Almost!
1. The example of the triangle shown to the right side is NOT the right triangle which is the trigonometric triangle. The triangle shown linearly (and incorrectly) demonstrates ratios only for the sin, sec and tan and displays them all wrong. That it is not the trigonometric triangle is evident in that a SECOND illustration is necessary to linearly display those ratios for the cos, cot and csc as linear expression of functions. NOT ONE TRIGONOMETRIC TRIANGLE, RATHER TWO ERRORS.
2. Modern definitions, and therefore the rote of modern mathematics, fail to recognize two very interesting and distinct functions which are quite present in the trigonometric triangle. Those functions are the dav and codav. They are so present in the trigonometric triangle they cannot be denied. One just needs to first achieve an understanding of the Trigonometric Triangle.
To do so see Google Videos
Search term: trigonometric triangle trilogy
or view (to make it easy) at these Google links:
(self-aggrandizing spam links removed)
Kindest regards, Keith Davie ( talk) 02:14, 12 May 2008 (UTC)
Amusing. Videos posted on YouTube.
"your dismissal of the function yielding the length of XY"
Prejudice is an ugly bedmate.
The functions which above are described as simple subtractions and additions in the linear geometric perform exponentially when studied through 91 degrees (0 through 90), hence sin(sq)/cos and cos(sq)/sin are more appropriate descriptions.
Regarding an 'extra layer of complexity", there is far more elegance in the Trigonometric Triangle of my video than either two triangles or a first quadrant angle with a triangle in the fourth quadrant, both products of Wiki. Imagine using one triangle to measure the angle. Oh my, too complex for Wiki.
And if one starts counting at angle no. 1, like the trig tables tell us, at zero degrees (it is not to be feared), when one arrives at 360 degrees that is angle 361. Good grief, I am far to radical to be taken seriously, claiming there are 361 degrees in a complete circle. Have a fun time with that at my expense guys. :)
Kindest regards Keith Davie ( talk) 01:14, 14 May 2008 (UTC) Keith Davie
Excuse me, Maelin, if you took offense, but I do not recall that I dismissed a "function yielding the length XY". I do appreciate the kind words you have offered. Perhaps you were referring to that which Black Carrot commented upon, and you made a typo intending XZ and not XY. And I forgive Black Carrot, who refuses to recognize these two functions and finds the Trigonometric Triangle only good for a way to fit all six of the ordinary trig functions into a single diagram. It's eight functions which fit in, with the versine and the exsec. as bonuses. Yeah, all the functions are there in one triangle in the first quadrant.
And while my understanding of what is a function may be somewhat wanting, I cannot find where I said, as Black Carrot seems to allege, that the angle is a function of XZ. My point is that the Trigonometric Triangle, i.e., the triangle which measures the functions, is constructed from the given angle in terms of (or perhaps in relation to) the unit circle.
And while the functions may repeat in the second, third and fourth quadrants, that could, with an added layer of complexity, be resolved by a functional notation for each quadrant. Each angle is unique in its input and an angle of 173 degrees does not occur in the first quadrant. Perhaps math has a notational issue to deal with. Oh my.
As to the argument, What mathematics (or, more accurately, mathematicians) hasn't done is give these functions special names, probably because they are not especially useful or interesting, it may also be that, given the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions. Given the examples of triangles mathematicians have used to measure the angle, I believe there is a stronger case to be made for the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions.
While it may appear that I have 'come off as a god of mathematics', I would suggest that the argument 'because they are not especially useful or interesting' raises the questions of to whom and isn't that denial of openness of knowledge itself acting as a god of mathematics. Obviously, if I am the pot then I am good company with the kettles.
It ain't about me folks. It's about elegance of solution and knowledge. While the exercise here is somewhat enjoyable, the videos are there and they will gain viewers. And as they do they will generate further discussion. And I am certain that in time the Trigonometric Triangle and the two functions that math has shunned for whatever reason will, of necessity, become accepted.
As to the added layer of complexity, I take it that concern is in regards to teaching minds which are less than inquiring and not in regards the communication of knowledge, which is what separates homo sapiens from all the other species.
Kindest regards, Keith Davie ( talk) 04:43, 14 May 2008 (UTC)Keith Davie
Yes, thank you for your advice. While it can appear that zero degrees and 360 degrees are the same angle, that is deception of appearance. Is the angle of 360 degrees a fourth quadrant angle? I know that the angle of zero degrees is where I am when I put the compass point and lead to paper. As to the two functions can be described by the 6 functions, I observe of the eight functions, six functions can be described by the quotients sin and cos. Let us say that such descriptions are theoretical. Similarly the eight functions can be described numerically. Similarly the eight functions can be described geometrically. What parts of that knowledge should we dismiss? Is it possible the rote typically described in math class is stifling inquiring minds? I don't know. I haven't been in a trig class in nearly half a century and I don't feel like my mind is stifled.
As to the dismissal of XZ as a function (mentioned above) I said, However, such a conclusion regarding XZ is an error. Perhaps the first clue is that XZ has no cofunction. As we observed earlier, the sine of one given angle is the cosine of that given angle's complementary angle, which allows for the input of each angle to result in a unique output for the function. However, as can be seen in this image, complementary angles have the same value for XZ, which lack of uniqueness precludes XZ from being a function.
The image shows complementary angles 27 - 32 and 63 - 58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles. For instance the values of the eight functions of 27 degrees is unique to those functions of 27 degrees. What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?
Please keep in mind regarding my initial query that while I took issue with the triangles which Wiki has chosen to display, I offered a solution. — Keith Davie ( talk) 13:43, 14 May 2008 (UTC)Keith Davie
Regarding the comment about 360 being the same as 0 degrees. Like I said, it does give that appearance but the appearance is deceptive. I than asked, isn't that a fourth quadrant angle.
Dance of the Trigonometric Tables
I awoke this morning thinking of the dance which occurs between the cotangent and tangent as an angle increases from 0 to 90 degrees, at least in the model ballroom of the Trigonometric Triangle which I provide. At 0 degrees the tangent's value is zero and the cotangent dances away from the tangent through the first quadrant to the eternity of the plane. As the angle progression occurs the tangent lengthens and the cotangent shortens. The cotangent and tangent are always in contact on the circumference at the point where the rising side of the angel, the radius, intersects. Finally the cotnagent arrives at the spot on the dance floor where its value is zero. Here the tangent dances away from the cotangent through the first quadrant to the eternity of the plane.
In my ballroom's model, when the dance continues into the second quadrant with the first hint of obtuseness, the cotangent begins to appear and the tangent performs a grand jeté into the second quadrant. And so their dance continues with the cotangent lengthening and the tangent shortening to 180 degrees where the tangent again is zero and the cotangent dances away from the tangent across the dancefloor of the second quadrant to the eternity of the plane.
And this dance repeats again into the third quadrant where the cotangent performs a grand geté and then into the fourth with the tangent again performing the grand jeté as it enters. And when the couple arrive at degree 359, the 360th angle of their Dance of the Trigonometric Tables, they have not completed the dance. There is one more angle before the dance is where it began. And when they take that 361st step the tangent is at zero but the cotangent is still in the fourth quadrant, dancing across its dancefloor to the eternity of the plane.
So while the appearance of the angle may look the same, the Trigonometric Triangle reveals a different image and the deception of appearance.
Keith Davie ( talk) 13:43, 14 May 2008 (UTC)Keith Davie
Dear responders,
Thank you for your responses. I have reviewed them with a bit of distance from the jostlings of the moment.
I see where one could take offense at my original bold statements.
Especially rankling seems to have been my statement that modern mathematics fails to recognize these two functions. Perhaps I could have been more gentle, but consider the two triangles currently displayed at Wiki. Neither mentions nor shows their existence. That is hardly an acknowledgment.
Similarly the first degree angle image which Kinu mentions does not mention these two functions, rather it shows the versine and exsecant, i.e., two functions where one will do. An odd complexity in my opinion. Especially since neither is defined by the sin and cos as the other functions, including Davie's. That image also has the problem of an unnecessary and less inelegant extension into the fourth quadrant. Where's the acknowledgment in that triangle?
Trig Tables? No acknowledgment. Nor did anyone post a reference which showed that the Trigonometric Triangle and the two functions I claim as Davie's functions had been considered and rejected by peer review.
And while T'hey are easy to define, they haven't been until now, hence Davie's Functions. It is also easy to mis-define them. It's like those tools do not exist for mathematicians. It should not be surprising then that the tools are not used.
As to the other issues raised by the responders, many were either errors such as typos and incorrect references or flat out misrepresentations of what was said by me in the videos and even on this board.
Concerning my dismissal of XZ as a function, I understand full well the argument that arises with 2nd, 3rd, and 4th quadrant sharing of values. If you watched all three of the videos you know that I addressed that issue as the usual considerations. If we don't make those considerations, then math has a notational issue. We can't have these things both ways.
In regards to first quadrant angles, the acute angles with which I was dealing, I am accurate in my statement and my mathematics apparently since no one has yet answered my responding query, What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?. To that I will add, where is it's co-function, and if it does not have one, what gives it that exception also?
The final thing which everyone seems to have missed is in the very first sentence throughout my video trilogy I say, ...I am not a mathematician. Instead of taking me to task for my inelegant verbal and written expressions perhaps one of you could have seen the opportunity which these tools presents to a mathematician to more elegantly and fully discuss these tools (Trigonometric Triangle model and Davie's Functions) than that which I am capable or trained to do. Good tools belong in the mathematics toolbox so their uses can be found or realized by others. What if Fibonacci had been poo-pooed because there was not Dow Jones so he could not have known his theory would be applied to that market by someone inspired by his work? If one is denied the tools, one will never use them.
I have considered the things which I put in my videos for over thirty years and am confident in the math and speak confidently of the Trigonometric Triangle model I presented and Davie's Functions, which are best defined through their theoretical exponential expression of sin^2/cos and cos^2/sin, for that is how they act in linear, geometric progressions over several angles. My model is based in part on the standard math tables and is in agreement with that.
I now have posted one more video at Google Video. Those interested will look. I am not posting the link so Wiki Math Reference Desk will not need delete it.
This is not about self-aggrandizement but about the dissemination of knowledge.
Now playing at Google is a silent movie called Function Cotillion (sub: Having A Ball With Geometry), presenting a set of dances displaying the actions of the eight functions as dance partners, geometrically expressed in the first quadrant.
It too might provide someone some inspiration some day.
I made an offering of a model when I came here. It has been rejected because it is boring. It appears a high point in mathematics has been achieved at Wiki.
Again, thank you all for your responses. Even the negatives have made me more confident in the mathematics and elegance of my model. Nothing better has been offered.
I have no regrets.
Kindest regards Keith Davie ( talk) 22:18, 16 May 2008 (UTC)Keith Davie
I regret to inform that the video I first uploaded to Google is not playing correctly, dropping a lot of data, nor did it download correctly. I have just uploaded a replacement, but it may take a while for the upload to take effect. I will check later. If that one does not work, I will try a very low-grade and blurry format as an upload. I will check my talk box later and if anyone is interested I will attempt to email you a copy if you tell me where. It runs in less than three minutes and is less than one meg. Keith Davie ( talk) 22:50, 16 May 2008 (UTC)Keith Davie
I am pleased to inform that the movie Function Cotillion has been uploaded successfully at last at Google Video. The QuickTime movie, which runs in just under three minutes, downloads adequately as well. I had to enlarge the video to ~3 mb.
Maelin, I have answered your at my talk.
Keith Davie ( talk) 00:03, 20 May 2008 (UTC)Keith Davie
The solutions to a particular equation can consist of different kinds of real numbers,for example rational numbers,or irrational;they can be equal roots or unequal,positive,negative or zero.There can be even no real solutions at all.All of this depends on the values of a,b and c in a given equation. The quantity under the square root(delta) controls what kind of roots there are for a given equation.For example if this quantity is negative then there are no rea roots.
QUESTION- What are the conditions that apply to delta, so that the solutions will fit into these categories-real,unreal,rational,irrational,aqual,unequal? —Preceding unsigned comment added by 196.207.40.212 ( talk) 14:23, 12 May 2008 (UTC)
This is a question about notation. Specifically I’m working in a noncommutative ring with nonequivalent left and right division algorithms. The question is if there is any accepted notation for saying that “a left divides c” or “b right divides c”. Clearly, “a|c” won’t work because it’s ambiguous if a doesn’t both left and right divide c. GromXXVII ( talk) 15:15, 12 May 2008 (UTC)
What is the connection between the definitions of the unit sphere and the real-projective plane? I was told they were connected. I have seen how the geodesics on the sphere correspond to lines in the real-projective plane, but when you just look at definitions of spaces I get all confused.. Also, how is it that the real-projective plane can be regarded as a disc? I can't find an explanation of this anywhere! —Preceding unsigned comment added by 152.78.120.74 ( talk) 15:26, 12 May 2008 (UTC)
A point in the real projective plane can be regarded as a pair of antipodal points on the sphere. Michael Hardy ( talk) 20:49, 15 May 2008 (UTC)
I have exams coming up in two weeks and this is a past paper question, i'm worried something similar will come up again this year.. can someone please explain? If you had 5 points in the real-projective plane, no three of which are collinear, how would you prove that there is a unique non-degenerate conic passing through each of the points? 152.78.120.74 ( talk) 15:40, 12 May 2008 (UTC)
I am trying to figure out how many permutations I can have of 4 values, where the values are integers from 0 to 100 inclusive. The four values sum to 100 and the position is important. I have figured out that if I have 1, 2, or 3 values, the answer is 1, 101, and 5050 (101 choose 2). How can I do the next step for the 4 value case? I figure this is a nested binomial coeffienct or choose 2 problem. (I realize a good upper bound is 101^4 ~ 10^8. I have calculated 2.5 x 10^5 as a close solution, but I'd like to be exact if possible.)
Also, I am thinking about a case with a larger number like 23. Where I have 23 integer values that sum to 100 inclusive. This would be a massive nested problem. Thanks for your help. -- Rajah ( talk) 22:22, 12 May 2008 (UTC)
Hi. These are not specific homework questions that I have to figure out, nor are they things I have to figure out by myself using some kind of formula, as I am asking for a way to measure these and not the answer to any questions or specific measurements (and am seeing if my formulas are correct). These questions are about making nets for constructing a shell of a geometric solid. First question I have is about the net of a right circular cone. I am to make a net so that the edges so not overlap and to only use tabs for connection. The net for a cone consists of a circle and a shape that is almost like a geograhpic isoceles triangle, where the congruent edges are straight but the edge connecting to the circle is an arc. I understand that the arc's length is equal to the circumfrence of the circle, but how do I draw the arc to the right curvature? I have the height of the cone as well as the diametre of the circle. If you extend the arc, what shape is it and how large is the resulting shape? I'm thinking that it will be a circle, with the centre of the circle at the vertex of the cone's net, as the distance from the vertex to the bottom of the shape which connects to the circle should remain the same, as this is a right cone and the distance on the curved face of the cone between the vertex and the circular edge should be the same. Am I correct, or is there a different simple formula for this? Thus, is the formula for the distances between the vertex and the round edge on the triangle-like portion of the cone's net [math] sqrt(r2+h2) [/math]? Sorry, I'm not very good with LaTeX syntax and r means the radius of the circle and h means the height of the cone. Ok, next I am to do a chevron-based pyramid. What I mean by that is, the base is a chevron shape, which is kind of like, in this case, an isoceles triangle with another triangle cut out of it at the only edge not congruent to the other two edges, that is also isoceles. So, the end result (chevron) should be a quadrilateral with three acute angles, and one reflex angle, and the reflex angle, were is subtracted from 360 to make an acute angle, would be greater than the angle that is pointing in the same direction and is not the two congruent acute angles. Basicly, I hope you know what a chevron is. The vertex for the pyramid is directly above the reflex angle, and the resulting pyramid should have 5 faces, 8 edges, and 5 verticies. However, that's not my question. In order to make a net, will I need to use any trigonometric functions (sine, cosine, tangent) to calculate the lengths of the edges for each of the 5 faces for the net, other than the basic Pathegorean theorem? I know the distance between the two congruent angles on the chevron (the width), the length between the other acute angle on the chevron and the reflex angle on the chevron (the length1), the distance between the reflex angle and the middle of the width line, which forms the edge of the nonexistant triangle in the chevron that does not share the edges with the real shape (the length2) [length1 and length2 should form a straight line with each other, as well as the height, which is the distance between the reflex angle and the top vertex of the pyramid. My other question is, is there a way to calculate whether or not a net with 6 congruent squares will succesfully form a cube if folded, without actually folding it? Something like x edges to seal, x edges to fold, etc. Thanks. ~ A H 1( T C U) 22:34, 12 May 2008 (UTC)
Mathematics desk | ||
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< May 11 | << Apr | May | Jun >> | May 13 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Per 6/49, the chances of winning the jackpot is roughly 1 in 14 million. However, lotto 6/49 consists of matching 6 numbers from a box of 49 numbers. So therefore, shouldn't the odds of winning the jackpot be 1 in (69 x 68 x 67 x 66 x 65 x 64) or 1 in 86311779840? Where have I gone wrong? Acceptable ( talk) 00:50, 12 May 2008 (UTC)
Oh ok, haha I don't know why i used 69 and 68's instead of 49 and 48's. Thanks. Acceptable ( talk) 22:28, 12 May 2008 (UTC)
The article at http://en.wikipedia.org/wiki/Trigonometric_function states, "Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers."
Almost!
1. The example of the triangle shown to the right side is NOT the right triangle which is the trigonometric triangle. The triangle shown linearly (and incorrectly) demonstrates ratios only for the sin, sec and tan and displays them all wrong. That it is not the trigonometric triangle is evident in that a SECOND illustration is necessary to linearly display those ratios for the cos, cot and csc as linear expression of functions. NOT ONE TRIGONOMETRIC TRIANGLE, RATHER TWO ERRORS.
2. Modern definitions, and therefore the rote of modern mathematics, fail to recognize two very interesting and distinct functions which are quite present in the trigonometric triangle. Those functions are the dav and codav. They are so present in the trigonometric triangle they cannot be denied. One just needs to first achieve an understanding of the Trigonometric Triangle.
To do so see Google Videos
Search term: trigonometric triangle trilogy
or view (to make it easy) at these Google links:
(self-aggrandizing spam links removed)
Kindest regards, Keith Davie ( talk) 02:14, 12 May 2008 (UTC)
Amusing. Videos posted on YouTube.
"your dismissal of the function yielding the length of XY"
Prejudice is an ugly bedmate.
The functions which above are described as simple subtractions and additions in the linear geometric perform exponentially when studied through 91 degrees (0 through 90), hence sin(sq)/cos and cos(sq)/sin are more appropriate descriptions.
Regarding an 'extra layer of complexity", there is far more elegance in the Trigonometric Triangle of my video than either two triangles or a first quadrant angle with a triangle in the fourth quadrant, both products of Wiki. Imagine using one triangle to measure the angle. Oh my, too complex for Wiki.
And if one starts counting at angle no. 1, like the trig tables tell us, at zero degrees (it is not to be feared), when one arrives at 360 degrees that is angle 361. Good grief, I am far to radical to be taken seriously, claiming there are 361 degrees in a complete circle. Have a fun time with that at my expense guys. :)
Kindest regards Keith Davie ( talk) 01:14, 14 May 2008 (UTC) Keith Davie
Excuse me, Maelin, if you took offense, but I do not recall that I dismissed a "function yielding the length XY". I do appreciate the kind words you have offered. Perhaps you were referring to that which Black Carrot commented upon, and you made a typo intending XZ and not XY. And I forgive Black Carrot, who refuses to recognize these two functions and finds the Trigonometric Triangle only good for a way to fit all six of the ordinary trig functions into a single diagram. It's eight functions which fit in, with the versine and the exsec. as bonuses. Yeah, all the functions are there in one triangle in the first quadrant.
And while my understanding of what is a function may be somewhat wanting, I cannot find where I said, as Black Carrot seems to allege, that the angle is a function of XZ. My point is that the Trigonometric Triangle, i.e., the triangle which measures the functions, is constructed from the given angle in terms of (or perhaps in relation to) the unit circle.
And while the functions may repeat in the second, third and fourth quadrants, that could, with an added layer of complexity, be resolved by a functional notation for each quadrant. Each angle is unique in its input and an angle of 173 degrees does not occur in the first quadrant. Perhaps math has a notational issue to deal with. Oh my.
As to the argument, What mathematics (or, more accurately, mathematicians) hasn't done is give these functions special names, probably because they are not especially useful or interesting, it may also be that, given the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions. Given the examples of triangles mathematicians have used to measure the angle, I believe there is a stronger case to be made for the triangles mathematicians have used to measure the angle have not been a triangle which encountered these distinct functions.
While it may appear that I have 'come off as a god of mathematics', I would suggest that the argument 'because they are not especially useful or interesting' raises the questions of to whom and isn't that denial of openness of knowledge itself acting as a god of mathematics. Obviously, if I am the pot then I am good company with the kettles.
It ain't about me folks. It's about elegance of solution and knowledge. While the exercise here is somewhat enjoyable, the videos are there and they will gain viewers. And as they do they will generate further discussion. And I am certain that in time the Trigonometric Triangle and the two functions that math has shunned for whatever reason will, of necessity, become accepted.
As to the added layer of complexity, I take it that concern is in regards to teaching minds which are less than inquiring and not in regards the communication of knowledge, which is what separates homo sapiens from all the other species.
Kindest regards, Keith Davie ( talk) 04:43, 14 May 2008 (UTC)Keith Davie
Yes, thank you for your advice. While it can appear that zero degrees and 360 degrees are the same angle, that is deception of appearance. Is the angle of 360 degrees a fourth quadrant angle? I know that the angle of zero degrees is where I am when I put the compass point and lead to paper. As to the two functions can be described by the 6 functions, I observe of the eight functions, six functions can be described by the quotients sin and cos. Let us say that such descriptions are theoretical. Similarly the eight functions can be described numerically. Similarly the eight functions can be described geometrically. What parts of that knowledge should we dismiss? Is it possible the rote typically described in math class is stifling inquiring minds? I don't know. I haven't been in a trig class in nearly half a century and I don't feel like my mind is stifled.
As to the dismissal of XZ as a function (mentioned above) I said, However, such a conclusion regarding XZ is an error. Perhaps the first clue is that XZ has no cofunction. As we observed earlier, the sine of one given angle is the cosine of that given angle's complementary angle, which allows for the input of each angle to result in a unique output for the function. However, as can be seen in this image, complementary angles have the same value for XZ, which lack of uniqueness precludes XZ from being a function.
The image shows complementary angles 27 - 32 and 63 - 58 respectively sharing the same values for XZ, Aside from the special case 45 degree angle which is its own complementary angle, other acute angles and their complementary angles share no value for the same function, keeping it simple here with first quadrant acute angles. For instance the values of the eight functions of 27 degrees is unique to those functions of 27 degrees. What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?
Please keep in mind regarding my initial query that while I took issue with the triangles which Wiki has chosen to display, I offered a solution. — Keith Davie ( talk) 13:43, 14 May 2008 (UTC)Keith Davie
Regarding the comment about 360 being the same as 0 degrees. Like I said, it does give that appearance but the appearance is deceptive. I than asked, isn't that a fourth quadrant angle.
Dance of the Trigonometric Tables
I awoke this morning thinking of the dance which occurs between the cotangent and tangent as an angle increases from 0 to 90 degrees, at least in the model ballroom of the Trigonometric Triangle which I provide. At 0 degrees the tangent's value is zero and the cotangent dances away from the tangent through the first quadrant to the eternity of the plane. As the angle progression occurs the tangent lengthens and the cotangent shortens. The cotangent and tangent are always in contact on the circumference at the point where the rising side of the angel, the radius, intersects. Finally the cotnagent arrives at the spot on the dance floor where its value is zero. Here the tangent dances away from the cotangent through the first quadrant to the eternity of the plane.
In my ballroom's model, when the dance continues into the second quadrant with the first hint of obtuseness, the cotangent begins to appear and the tangent performs a grand jeté into the second quadrant. And so their dance continues with the cotangent lengthening and the tangent shortening to 180 degrees where the tangent again is zero and the cotangent dances away from the tangent across the dancefloor of the second quadrant to the eternity of the plane.
And this dance repeats again into the third quadrant where the cotangent performs a grand geté and then into the fourth with the tangent again performing the grand jeté as it enters. And when the couple arrive at degree 359, the 360th angle of their Dance of the Trigonometric Tables, they have not completed the dance. There is one more angle before the dance is where it began. And when they take that 361st step the tangent is at zero but the cotangent is still in the fourth quadrant, dancing across its dancefloor to the eternity of the plane.
So while the appearance of the angle may look the same, the Trigonometric Triangle reveals a different image and the deception of appearance.
Keith Davie ( talk) 13:43, 14 May 2008 (UTC)Keith Davie
Dear responders,
Thank you for your responses. I have reviewed them with a bit of distance from the jostlings of the moment.
I see where one could take offense at my original bold statements.
Especially rankling seems to have been my statement that modern mathematics fails to recognize these two functions. Perhaps I could have been more gentle, but consider the two triangles currently displayed at Wiki. Neither mentions nor shows their existence. That is hardly an acknowledgment.
Similarly the first degree angle image which Kinu mentions does not mention these two functions, rather it shows the versine and exsecant, i.e., two functions where one will do. An odd complexity in my opinion. Especially since neither is defined by the sin and cos as the other functions, including Davie's. That image also has the problem of an unnecessary and less inelegant extension into the fourth quadrant. Where's the acknowledgment in that triangle?
Trig Tables? No acknowledgment. Nor did anyone post a reference which showed that the Trigonometric Triangle and the two functions I claim as Davie's functions had been considered and rejected by peer review.
And while T'hey are easy to define, they haven't been until now, hence Davie's Functions. It is also easy to mis-define them. It's like those tools do not exist for mathematicians. It should not be surprising then that the tools are not used.
As to the other issues raised by the responders, many were either errors such as typos and incorrect references or flat out misrepresentations of what was said by me in the videos and even on this board.
Concerning my dismissal of XZ as a function, I understand full well the argument that arises with 2nd, 3rd, and 4th quadrant sharing of values. If you watched all three of the videos you know that I addressed that issue as the usual considerations. If we don't make those considerations, then math has a notational issue. We can't have these things both ways.
In regards to first quadrant angles, the acute angles with which I was dealing, I am accurate in my statement and my mathematics apparently since no one has yet answered my responding query, What would cause an exception for XZ as a function of a unique angle to provide output which is not unique?. To that I will add, where is it's co-function, and if it does not have one, what gives it that exception also?
The final thing which everyone seems to have missed is in the very first sentence throughout my video trilogy I say, ...I am not a mathematician. Instead of taking me to task for my inelegant verbal and written expressions perhaps one of you could have seen the opportunity which these tools presents to a mathematician to more elegantly and fully discuss these tools (Trigonometric Triangle model and Davie's Functions) than that which I am capable or trained to do. Good tools belong in the mathematics toolbox so their uses can be found or realized by others. What if Fibonacci had been poo-pooed because there was not Dow Jones so he could not have known his theory would be applied to that market by someone inspired by his work? If one is denied the tools, one will never use them.
I have considered the things which I put in my videos for over thirty years and am confident in the math and speak confidently of the Trigonometric Triangle model I presented and Davie's Functions, which are best defined through their theoretical exponential expression of sin^2/cos and cos^2/sin, for that is how they act in linear, geometric progressions over several angles. My model is based in part on the standard math tables and is in agreement with that.
I now have posted one more video at Google Video. Those interested will look. I am not posting the link so Wiki Math Reference Desk will not need delete it.
This is not about self-aggrandizement but about the dissemination of knowledge.
Now playing at Google is a silent movie called Function Cotillion (sub: Having A Ball With Geometry), presenting a set of dances displaying the actions of the eight functions as dance partners, geometrically expressed in the first quadrant.
It too might provide someone some inspiration some day.
I made an offering of a model when I came here. It has been rejected because it is boring. It appears a high point in mathematics has been achieved at Wiki.
Again, thank you all for your responses. Even the negatives have made me more confident in the mathematics and elegance of my model. Nothing better has been offered.
I have no regrets.
Kindest regards Keith Davie ( talk) 22:18, 16 May 2008 (UTC)Keith Davie
I regret to inform that the video I first uploaded to Google is not playing correctly, dropping a lot of data, nor did it download correctly. I have just uploaded a replacement, but it may take a while for the upload to take effect. I will check later. If that one does not work, I will try a very low-grade and blurry format as an upload. I will check my talk box later and if anyone is interested I will attempt to email you a copy if you tell me where. It runs in less than three minutes and is less than one meg. Keith Davie ( talk) 22:50, 16 May 2008 (UTC)Keith Davie
I am pleased to inform that the movie Function Cotillion has been uploaded successfully at last at Google Video. The QuickTime movie, which runs in just under three minutes, downloads adequately as well. I had to enlarge the video to ~3 mb.
Maelin, I have answered your at my talk.
Keith Davie ( talk) 00:03, 20 May 2008 (UTC)Keith Davie
The solutions to a particular equation can consist of different kinds of real numbers,for example rational numbers,or irrational;they can be equal roots or unequal,positive,negative or zero.There can be even no real solutions at all.All of this depends on the values of a,b and c in a given equation. The quantity under the square root(delta) controls what kind of roots there are for a given equation.For example if this quantity is negative then there are no rea roots.
QUESTION- What are the conditions that apply to delta, so that the solutions will fit into these categories-real,unreal,rational,irrational,aqual,unequal? —Preceding unsigned comment added by 196.207.40.212 ( talk) 14:23, 12 May 2008 (UTC)
This is a question about notation. Specifically I’m working in a noncommutative ring with nonequivalent left and right division algorithms. The question is if there is any accepted notation for saying that “a left divides c” or “b right divides c”. Clearly, “a|c” won’t work because it’s ambiguous if a doesn’t both left and right divide c. GromXXVII ( talk) 15:15, 12 May 2008 (UTC)
What is the connection between the definitions of the unit sphere and the real-projective plane? I was told they were connected. I have seen how the geodesics on the sphere correspond to lines in the real-projective plane, but when you just look at definitions of spaces I get all confused.. Also, how is it that the real-projective plane can be regarded as a disc? I can't find an explanation of this anywhere! —Preceding unsigned comment added by 152.78.120.74 ( talk) 15:26, 12 May 2008 (UTC)
A point in the real projective plane can be regarded as a pair of antipodal points on the sphere. Michael Hardy ( talk) 20:49, 15 May 2008 (UTC)
I have exams coming up in two weeks and this is a past paper question, i'm worried something similar will come up again this year.. can someone please explain? If you had 5 points in the real-projective plane, no three of which are collinear, how would you prove that there is a unique non-degenerate conic passing through each of the points? 152.78.120.74 ( talk) 15:40, 12 May 2008 (UTC)
I am trying to figure out how many permutations I can have of 4 values, where the values are integers from 0 to 100 inclusive. The four values sum to 100 and the position is important. I have figured out that if I have 1, 2, or 3 values, the answer is 1, 101, and 5050 (101 choose 2). How can I do the next step for the 4 value case? I figure this is a nested binomial coeffienct or choose 2 problem. (I realize a good upper bound is 101^4 ~ 10^8. I have calculated 2.5 x 10^5 as a close solution, but I'd like to be exact if possible.)
Also, I am thinking about a case with a larger number like 23. Where I have 23 integer values that sum to 100 inclusive. This would be a massive nested problem. Thanks for your help. -- Rajah ( talk) 22:22, 12 May 2008 (UTC)
Hi. These are not specific homework questions that I have to figure out, nor are they things I have to figure out by myself using some kind of formula, as I am asking for a way to measure these and not the answer to any questions or specific measurements (and am seeing if my formulas are correct). These questions are about making nets for constructing a shell of a geometric solid. First question I have is about the net of a right circular cone. I am to make a net so that the edges so not overlap and to only use tabs for connection. The net for a cone consists of a circle and a shape that is almost like a geograhpic isoceles triangle, where the congruent edges are straight but the edge connecting to the circle is an arc. I understand that the arc's length is equal to the circumfrence of the circle, but how do I draw the arc to the right curvature? I have the height of the cone as well as the diametre of the circle. If you extend the arc, what shape is it and how large is the resulting shape? I'm thinking that it will be a circle, with the centre of the circle at the vertex of the cone's net, as the distance from the vertex to the bottom of the shape which connects to the circle should remain the same, as this is a right cone and the distance on the curved face of the cone between the vertex and the circular edge should be the same. Am I correct, or is there a different simple formula for this? Thus, is the formula for the distances between the vertex and the round edge on the triangle-like portion of the cone's net [math] sqrt(r2+h2) [/math]? Sorry, I'm not very good with LaTeX syntax and r means the radius of the circle and h means the height of the cone. Ok, next I am to do a chevron-based pyramid. What I mean by that is, the base is a chevron shape, which is kind of like, in this case, an isoceles triangle with another triangle cut out of it at the only edge not congruent to the other two edges, that is also isoceles. So, the end result (chevron) should be a quadrilateral with three acute angles, and one reflex angle, and the reflex angle, were is subtracted from 360 to make an acute angle, would be greater than the angle that is pointing in the same direction and is not the two congruent acute angles. Basicly, I hope you know what a chevron is. The vertex for the pyramid is directly above the reflex angle, and the resulting pyramid should have 5 faces, 8 edges, and 5 verticies. However, that's not my question. In order to make a net, will I need to use any trigonometric functions (sine, cosine, tangent) to calculate the lengths of the edges for each of the 5 faces for the net, other than the basic Pathegorean theorem? I know the distance between the two congruent angles on the chevron (the width), the length between the other acute angle on the chevron and the reflex angle on the chevron (the length1), the distance between the reflex angle and the middle of the width line, which forms the edge of the nonexistant triangle in the chevron that does not share the edges with the real shape (the length2) [length1 and length2 should form a straight line with each other, as well as the height, which is the distance between the reflex angle and the top vertex of the pyramid. My other question is, is there a way to calculate whether or not a net with 6 congruent squares will succesfully form a cube if folded, without actually folding it? Something like x edges to seal, x edges to fold, etc. Thanks. ~ A H 1( T C U) 22:34, 12 May 2008 (UTC)