DimaMazin 93 Posted February 29, 2020 Share Posted February 29, 2020 I didn't like approximate definitions of trigonometric functions (it was about 34 years ago). Then I made speculation that side of angle (if angle is less or equal to Pi/2 rad) proportionally divides arc Pi/2 and its chord (2^{1/2}). Then my math teacher corrected me with her speculation that there is especial arc in which if to connect any two points of proportional division of this arc and its chord by straight line and to connect any two points of any another proportional division of this arc and its chord by another straight line , then the straight lines cross in one point of definition of trigonometric functions and angles(arcs). Is there any prize for exact definition of trigonometric functions and angles ? Link to post Share on other sites

studiot 2224 Posted February 29, 2020 Share Posted February 29, 2020 Your Google translated definitions seem very oddly phrased and over complicated. Can you explain simply what you are trying to do? All trigonometric functions are already very well defined. Link to post Share on other sites

DimaMazin 93 Posted March 1, 2020 Author Share Posted March 1, 2020 6 hours ago, studiot said: Your Google translated definitions seem very oddly phrased and over complicated. Can you explain simply what you are trying to do? All trigonometric functions are already very well defined. What is exact sin(Pi/4)=2^{1/2}/2 or sin(Pi/4)=0.7071067812 ? Can you define sine of any angle with exponents? Link to post Share on other sites

Strange 4273 Posted March 1, 2020 Share Posted March 1, 2020 8 hours ago, DimaMazin said: What is exact sin(Pi/4)=2^{1/2}/2 or sin(Pi/4)=0.7071067812 ? The first one is the exact value. The second is, obviously, an approximation. But I thought your question was about the definition of trig functions, not the value for a specific angle. So what do you actually want to know? (Your first post was rambling and incomprehensible; try splitting it up into a series of short simple statements, followed by a single question. And maybe a diagram.) 8 hours ago, DimaMazin said: Can you define sine of any angle with exponents? There are many ways of defining the sine function. Some are given here: https://en.m.wikipedia.org/wiki/Sine (That is probably also available in your favourite language.) Link to post Share on other sites

DimaMazin 93 Posted March 1, 2020 Author Share Posted March 1, 2020 1 hour ago, Strange said: The first one is the exact value. The second is, obviously, an approximation. But I thought your question was about the definition of trig functions, not the value for a specific angle. So what do you actually want to know? (Your first post was rambling and incomprehensible; try splitting it up into a series of short simple statements, followed by a single question. And maybe a diagram.) There are many ways of defining the sine function. Some are given here: https://en.m.wikipedia.org/wiki/Sine (That is probably also available in your favourite language.) Correct definition of trigonometric functions should make exact value for any specific angle. Concretely you don't need such definition. Thank you for honest answer. Link to post Share on other sites

Strange 4273 Posted March 1, 2020 Share Posted March 1, 2020 41 minutes ago, DimaMazin said: Correct definition of trigonometric functions should make exact value for any specific angle. And they do (whichever definition you use). So what is your question, or the point you are trying to make? 42 minutes ago, DimaMazin said: Concretely you don't need such definition. Well, obviously you do need the definition otherwise you could not calculate the value. So what is your question, or the point you are trying to make? Link to post Share on other sites

Ghideon 465 Posted March 1, 2020 Share Posted March 1, 2020 42 minutes ago, DimaMazin said: Correct definition of trigonometric functions should make exact value for any specific angle. Why? AFAIK value of trigonometric functions are exactly defined. Ability to numerically express a value exactly, using a limited amount of decimals, is not an issue with the definitions. Analogy: pi, defined as circumreference/diameter of a circle, seems to have an non ending sequence of decimals when expressed in base-10: 3.1415... that does not mean that circles, pi or their definitions are approximations. Link to post Share on other sites

taeto 93 Posted March 1, 2020 Share Posted March 1, 2020 (edited) 2 hours ago, Strange said: There are many ways of defining the sine function. Some are given here: https://en.m.wikipedia.org/wiki/Sine Maybe it also helps to answer the OP's question to observe that the link given by Strange refers to a basic definition of an angle as a union of two rays that meet in a point. There is nothing about real numbers or approximations. The \(\sin\) function is then defined by assigning a congruency class of line segments to each congruency class of angles. Other explanations, more popular nowadays, are derived by assigning real-valued measures to angles and line segments. But the original definition needs none of that. Edited March 1, 2020 by taeto Link to post Share on other sites

uncool 231 Posted March 1, 2020 Share Posted March 1, 2020 13 hours ago, DimaMazin said: What is exact sin(Pi/4)=2^{1/2}/2 or sin(Pi/4)=0.7071067812 ? Can you define sine of any angle with exponents? Depends on what you mean. For any angle A, sin(A) = (e^(i*A) - e^(-i*A))/2i, which is an expression of the sine using exponents, but I'm guessing that's not what you're going for (in part, because you seem to have an objection to numerically finding the value). It sounds like you are trying to define sines in terms of radicals of rational numbers. There are uncountably many angles, and only countably many expressions using radicals. So we'd have to restrict ourselves to some countable subset of angles. You chose to look at pi/4; that suggests using only rational multiples of pi. And with that restriction, the answer is yes - sin(pi*(a/b)) can always be expressed in terms of radicals. In fact, the expression I gave above can count: sin(pi*(a/b)) = (1^(a/2b) - 1^(-a/2b))/(2*(-1^(1/2))). That expression...isn't really helpful in finding the value of sin, but I'm pretty sure it can be converted to an expression that is. Link to post Share on other sites

DimaMazin 93 Posted March 1, 2020 Author Share Posted March 1, 2020 4 hours ago, Ghideon said: Why? AFAIK value of trigonometric functions are exactly defined. Ability to numerically express a value exactly, using a limited amount of decimals, is not an issue with the definitions. Analogy: pi, defined as circumreference/diameter of a circle, seems to have an non ending sequence of decimals when expressed in base-10: 3.1415... that does not mean that circles, pi or their definitions are approximations. Sine and cosine are coordinates of concrete point. They cannot be approximate. Otherwise how can you define equation of straight line which crosses this point? Link to post Share on other sites

Ghideon 465 Posted March 1, 2020 Share Posted March 1, 2020 18 minutes ago, DimaMazin said: Sine and cosine are coordinates of concrete point. They cannot be approximate. Otherwise how can you define equation of straight line which crosses this point? Can you clarify your question, maybe by providing a picture and/or formulas? Link to post Share on other sites

DimaMazin 93 Posted March 1, 2020 Author Share Posted March 1, 2020 14 minutes ago, Ghideon said: Can you clarify your question, maybe by providing a picture and/or formulas? I should not louse time. I should define arc of definition. Link to post Share on other sites

Ghideon 465 Posted March 1, 2020 Share Posted March 1, 2020 20 minutes ago, DimaMazin said: I should define arc of definition. Ok. Please post the result and the issue it solvets once you have made progress. In the meantime I’ll continue doing my work, using trigonometry, transforms, straight lines, curves and other mathematical concepts. From what I’ve learned so far there is no need for any new definition of arc or trigonometry. 25 minutes ago, DimaMazin said: I should not louse time Sorry that you think your own question is a waste of time. (Or maybe translation messed up your comment?) Link to post Share on other sites

Strange 4273 Posted March 1, 2020 Share Posted March 1, 2020 34 minutes ago, DimaMazin said: I should define arc of definition. What does "arc of definition" mean? Do you mean "definition of [an] arc" ? Link to post Share on other sites

studiot 2224 Posted March 1, 2020 Share Posted March 1, 2020 (edited) I still don't know what you want ? There are many 'definitions' of sinx and cos x. Obviously they all define the same things. So do you want me to tell you what sin and cos are? Or do you want a single formula that will calculate the value of sin or cos for any x. Remembering this value will only be 'correct' to a specific number of digits? Or do want ways of obtaining exact numeric values for any angle.? Remembering there is no single way to do this for every angle. Here are some single formulae in the form of continued products which converge at or before infinity. [math]\sin x = \left( {1 + \frac{x}{\pi }} \right)\left( {1 - \frac{x}{\pi }} \right)\left( {1 + \frac{x}{{2\pi }}} \right)\left( {1 - \frac{x}{{2\pi }}} \right)\left( {1 + \frac{x}{{3\pi }}} \right)\left( {1 - \frac{x}{{3\pi }}} \right)...[/math] [math]\cos x = \left( {1 + \frac{{2x}}{\pi }} \right)\left( {1 - \frac{{2x}}{\pi }} \right)\left( {1 + \frac{{2x}}{{3\pi }}} \right)\left( {1 - \frac{{2x}}{{3\pi }}} \right)\left( {1 + \frac{{2x}}{{5\pi }}} \right)\left( {1 - \frac{{2x}}{{5\pi }}} \right)...[/math] Edited March 1, 2020 by studiot Link to post Share on other sites

Strange 4273 Posted March 1, 2020 Share Posted March 1, 2020 3 hours ago, DimaMazin said: I should define arc of definition. I just tried Google translate to see if "arc of definition" meant something difference when translated into Russian. It doesn't seem to. So I have no idea what this means. Are you thinking in terms of an arc (presumably one subtended by the angle in question) that can be used to define the sine (or other) function? Link to post Share on other sites

druS 16 Posted March 2, 2020 Share Posted March 2, 2020 8 hours ago, DimaMazin said: Sine and cosine are coordinates of concrete point. They cannot be approximate. Otherwise how can you define equation of straight line which crosses this point? Out of interest to the greater wisdom here - I would not expect that sin of all angles can be defined exactly. Perhaps using limits/infetesimals? Link to post Share on other sites

DimaMazin 93 Posted March 2, 2020 Author Share Posted March 2, 2020 6 hours ago, Strange said: I just tried Google translate to see if "arc of definition" meant something difference when translated into Russian. It doesn't seem to. So I have no idea what this means. Are you thinking in terms of an arc (presumably one subtended by the angle in question) that can be used to define the sine (or other) function? Arc is angle(rad). I made some explore. If it exists then rather it is between 5/6 Pi and 17/18 Pi . Link to post Share on other sites

Strange 4273 Posted March 2, 2020 Share Posted March 2, 2020 3 hours ago, DimaMazin said: Arc is angle(rad). I made some explore. If it exists then rather it is between 5/6 Pi and 17/18 Pi . What is between 5/6 Pi and 17/18 Pi? Link to post Share on other sites

studiot 2224 Posted March 2, 2020 Share Posted March 2, 2020 14 hours ago, Strange said: What does "arc of definition" mean? Do you mean "definition of [an] arc" ? I think Dima is referring to the description of sin and cos as 'circular trigonometric functions' and their relationship to a rotating radius vector, as opposed to hyperbolic trigonometric functions. This is also linked to the use of angles to define sides of spherical triangles. But it would be nice to have a better answer to my question since it is a lot of work to investigate each aspect in depth. Link to post Share on other sites

DimaMazin 93 Posted March 2, 2020 Author Share Posted March 2, 2020 22 minutes ago, studiot said: I think Dima is referring to the description of sin and cos as 'circular trigonometric functions' and their relationship to a rotating radius vector, as opposed to hyperbolic trigonometric functions. This is also linked to the use of angles to define sides of spherical triangles. But it would be nice to have a better answer to my question since it is a lot of work to investigate each aspect in depth. Draw unit circle x^{2}+y^{2}=1 Mark angle 166 degrees or 168 degrees . Divide this angle for 6 equal parts. Draw chord of this arc(angle). Divide the chord for 6 equal parts. Draw straight line through 2 points , one of which divides arc for 1/6 part and 5/6 parts, another divides chord for 1/6 part and 5/6 parts. Then draw second straight line through 2 next points, one of which divides arc for 2/6 and 4/6 parts, another divides chord for 2/6 and 4/6 parts. Draw next straight lines through corresponding next points. If your arc is arc of definition then all these straight lines cross in one point of definition trigonometric functions and angles. If you know angle then you can define sine&cosine. If you know sine&cosine then you can define angle because you know coordinates of point of cross straight lines . Link to post Share on other sites

Country Boy 69 Posted March 2, 2020 Share Posted March 2, 2020 4 hours ago, Strange said: What is between 5/6 Pi and 17/18 Pi? 8/9 Pi is! Link to post Share on other sites

taeto 93 Posted March 2, 2020 Share Posted March 2, 2020 2 hours ago, Country Boy said: 8/9 Pi is! Well, yes, well done! The OP seems to have in mind some kind of riddle, to which the solution is an angle between \(5\pi /6\) and \(17\pi /18.\): ...there is especial arc in which if to connect any two points of proportional division of this arc and its chord by straight line and to connect any two points of any another proportional division of this arc and its chord by another straight line , then the straight lines cross in one point of definition of trigonometric functions and angles(arcs). Scratching my head to figure out what it means. Link to post Share on other sites

DimaMazin 93 Posted March 2, 2020 Author Share Posted March 2, 2020 4 hours ago, taeto said: Well, yes, well done! The OP seems to have in mind some kind of riddle, to which the solution is an angle between 5π/6 and 17π/18. : ...there is especial arc in which if to connect any two points of proportional division of this arc and its chord by straight line and to connect any two points of any another proportional division of this arc and its chord by another straight line , then the straight lines cross in one point of definition of trigonometric functions and angles(arcs). Scratching my head to figure out what it means. Yes and proportional divisions can be any , but coordinates of cross point should be constant. Link to post Share on other sites

taeto 93 Posted March 2, 2020 Share Posted March 2, 2020 10 minutes ago, DimaMazin said: Yes and proportional divisions can be any , but coordinates of cross point should be constant. What do you mean by an "arc" when you say "there is especial arc"? An arc can mean a piece of the unit circle, or just an angle, in the context. Like, there is an arc from \(0\) to \(\pi /2\) and an arc from \(\pi\) to \(3\pi /2,\) or do you think of them as just the same arc, of length \(\pi /2\)? Link to post Share on other sites

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