# How to define arc of definition?

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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 (21/2). 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 ?

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Correct. Divide arc for infinite quantity of equal parts and divide its chord for infinite quantity of equal parts. Draw straight line through  nearest point of division of the arc  to middle of the a

It may be easier and more efficient if you also provide a picture? If each interested member have to interpret your description there is plenty of room for confusion?

Edited by taeto
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13 hours ago, DimaMazin said:

What is exact  sin(Pi/4)=21/2/2 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.

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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?

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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?

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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.

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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?)

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34 minutes ago, DimaMazin said:

I should define arc of definition.

What does "arc of definition" mean?

Do you mean "definition of [an] arc" ?

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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.

$\sin x = \left( {1 + \frac{x}{\pi }} \right)\left( {1 - \frac{x}{\pi }} \right)\left( {1 + \frac{x}{{2\pi }}} \right)\left( {1 - \frac{x}{{2\pi }}} \right)\left( {1 + \frac{x}{{3\pi }}} \right)\left( {1 - \frac{x}{{3\pi }}} \right)...$

$\cos x = \left( {1 + \frac{{2x}}{\pi }} \right)\left( {1 - \frac{{2x}}{\pi }} \right)\left( {1 + \frac{{2x}}{{3\pi }}} \right)\left( {1 - \frac{{2x}}{{3\pi }}} \right)\left( {1 + \frac{{2x}}{{5\pi }}} \right)\left( {1 - \frac{{2x}}{{5\pi }}} \right)...$

Edited by studiot
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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?

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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?

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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 .

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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?

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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.

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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 x2+y2=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 .

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4 hours ago, Strange said:

What is between 5/6 Pi and 17/18 Pi?

8/9 Pi is!

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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.

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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.

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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 $$\pi /2$$?

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