DimaMazin

2 hours ago, taeto said:

That is impossible, since there is no nearest point. It would be like asking for the smallest positive number.

Then divide them for 1000¹⁰⁰⁰  equal parts. We don't need see it because we can use similar increasing of cathetuses of triangles (they increased are a/2 and chord/2) . I told about them.

13 hours ago, taeto said:

Then for every a≠0 you can calculate an x and a y. Is there anything wrong with that?

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 arc and through nearest point of division of the chord to middle of the chord, Draw straight line though middle point of the arc and middle point of the chord. Point of cross of the lines has coordinates (x ; y}.  Instead of first line you can draw next line: draw tangent in point (cos(a/2) ; sin(a/2)), mark length a/2  from point (cos(a/2) ; sin(a/2)) on the tangent and draw straight line through marked point and point (1: 0). Point of cross of new line and second line has coordinates (x ; y).    Marked point has coordinates (x₁ ; y₁)

x₁ = a*sin(a/2)/2+cos(a/2)

y₁ = sin²(a/2) - a*sin(a/2)*cos(a/2)/2

Yet we should draw straight line through nearest point of division the arc to edge and nearest point of division of the chord to edge. Then

add  half of the chord to point (1 ; 0) on straight line of the chord . On edge of the half of chord draw perpendicular to axis OX and mark length a/2 by point with coordinates(x₂ ; y₂)

x₂ =( 3 - cos(a))/2

y₂ = (a - sin(a))/2

If you draw straight line through points (x₂ ; y₂) and (1 ; 0) then the line crosses so called second line in point (x ; y).

If the arc is the arc of definition then coordinates (x ; y) are the same in both cases.But it is simpler to use equations of the straight lines.

Both the  lines have point(1;0) therefore in equation y=kx+b           b = - k

We define k₁ for one of the lines and k₂ for another of the lines and create equation of the arc of definition:    k₁ = k₂

On 3/14/2020 at 9:42 PM, Strange said:

 

I assume that is because they are wrong.

 

Yes. There:

x = cos(a/2)*(a*cos(a/2) - 2sin(a/2)) / (a - 2sin(a/2))

y = sin(a/2)*(a*cos(a/2) - 2sin(a/2)) / (a - 2sin(a/2))

19 hours ago, studiot said:

No.

I think perhaps there really is a language difficulty here.

I don't know how they teach geometry in Russia, but 'similar' is another technical word in Mathematics that has a special meaning,
This meaning relates particularly to shape.

(basic) geometry distinguishes 'lines' (as straight lines) and curves which are ''line' that are not straight.

An arc is part of (a segment of) a curve; a line segment is part of a line.

Therefore shape of circular arc is angle of its segment.

9 hours ago, studiot said:

 

Circular arcs have only one shape, by definition of circular.

 

Let's consider arc 2Pi and arc close to zero. Do you think circle and straight line have similar shapes?

27 minutes ago, studiot said:

@DimaMazin I am still waiting for an answer.

Shape of arc is feature of arc, but when we know radius we can use length as shape. 1 rad is complex angle because it has complex ratio to Pi.

My formulas show some problems with ratio to Pi .And I don't know will the method solve them or not.

If even the arc of definition doesn't define anything, I don't understand why it should be unknown? 

On 3/2/2020 at 3:18 PM, Country Boy said:

8/9 Pi is!

If it is correct 

(1 - cos(a))*sin(a/2)*(2sin(a/2) - a*cos(a/2)) / [(a - sin(a))*(a*sin(a/2)+2cos(a/2)  - 2)] = 1

Then you can check your angle.

Coordinates of definition point:

x = cos(a/2)*(a*cos(a/2) - 2sin(a/2)) / (2sin(a/2) - a)

y = sin(a/2)*(a*cos(a/2) - 2sin(a/2)) / (2sin(a/2) - a)

I don't know but maybe the method will help to define exact value of Pi .

On 3/5/2020 at 10:36 AM, DimaMazin said:

Coordinates of point of definition:

x = ( a - sin(a))*cos(a/2) / [cos(a/2)*(a - sin(a)) - sin(a/2)*(1 - cos(a))]

y = sin(a/2)*(a - sin(a)) / [cos(a/2)*(a - sin(a)) - sin(a/2)*(1 - cos(a))]

Pi/(2a) = sin(a/2)*(a - sin(a)+1 - cos(a))/[sin(a)*cos(a/2)*(a - sin(a))+(1 - cos(a))*((sin(a/2) - cos(a/2)*(a - sin(a))+sin(a/2)(1 - cos(a)))] 

Seems that is wrong. I should remake it.

2 hours ago, swansont said:

As opposed to what other systems?

Compare North Korea and USA. 

Everything is ineffective in socialism therefore military development is the most important.

1 hour ago, studiot said:

 

You are not really defining anything at all, and should not be using that word.

 

Rather the arc(angle) is very complex and its sine and cosine are more complex, only therefore we can not use this method for definition of trigonometric functions.

30 minutes ago, Ghideon said:

Question: By ”arc of definition” Do you mean ”an arc of the unit circle”?

Yes. If you know what is arcsine and arccosine then you should understand what is arc of unit circle. Arc of definition is part of unit circle.

Coordinates of point of definition:

x = ( a - sin(a))*cos(a/2) / [cos(a/2)*(a - sin(a)) - sin(a/2)*(1 - cos(a))]

y = sin(a/2)*(a - sin(a)) / [cos(a/2)*(a - sin(a)) - sin(a/2)*(1 - cos(a))]

1 hour ago, DimaMazin said:

Excuse me Studiot. I just showed how it can define unknown angle when arc of definition is known and coordinates of point of definition are known. But if your computer can solve very complex equation then please solve this one:

a is arc of definition ( rad)

Pi/(2a) = sin(a/2)*(a - sin(a)+1 - cos(a))/[sin(a)*cos(a/2)*(a - sin(a))+(1 - cos(a))*((sin(a/2) - cos(a/2)*(a - sin(a))+sin(a/2)(1 - cos(a))] 

Pi/(2a) = sin(a/2)*(a - sin(a)+1 - cos(a))/[sin(a)*cos(a/2)*(a - sin(a))+(1 - cos(a))*((sin(a/2) - cos(a/2)*(a - sin(a))+sin(a/2)(1 - cos(a)))] 

On 3/3/2020 at 11:38 AM, studiot said:

Thank you for your reply.

This is exactly what is puzzling myself and other members.

You consistently speak of 'definition', but you are using quantities you are trying to define in your definition.

(A computer would return a 'reference to undefined quantity' error)

I haven't yet checked your algebra for consistently - that will take time.
But you can't use something (eg the sine function) to define itself.

Excuse me Studiot. I just showed how it can define unknown angle when arc of definition is known and coordinates of point of definition are known. But if your computer can solve very complex equation then please solve this one:

a is arc of definition ( rad)

Pi/(2a) = sin(a/2)*(a - sin(a)+1 - cos(a))/[sin(a)*cos(a/2)*(a - sin(a))+(1 - cos(a))*((sin(a/2) - cos(a/2)*(a - sin(a))+sin(a/2)(1 - cos(a))] 

7 minutes ago, studiot said:

Thank you for your reply.

This is exactly what is puzzling myself and other members.

You consistently speak of 'definition', but you are using quantities you are trying to define in your definition.

(A computer would return a 'reference to undefined quantity' error)

I haven't yet checked your algebra for consistently - that will take time.
But you can't use something (eg the sine function) to define itself.

Sine is known there. I defined angle.

Length of arc of definition is a

coordinates of point of definition(cross point of 5 lines) are (x₁ ; y₁)

Length of chord is 2sin(a/2)

Equation of chord is     y=(sin(a)*x - sin(a))/(cos(a) - 1)

For example we know sin and cos of unknown angle 

Let's define unknown angle : coordinates of point of unknown angle on arc are (cos;sin)

Then draw straight line through points (cos;sin) and (x₁;y₁)   Equation of the line is:

y=(y₁ - sin)*x/(x₁ - cos)+sin - (y₁ - sin)*cos/(x₁ - cos)

Cross point of the line and chord has coordinates (x ' ;  y' )

We know equations of the line and  the chord therefore we can define x' (it is complex)

(1 - x')/(1 - cos(a))= part of divided chord / chord = unknown angle / a

unknown angle = a*(1 - x')/(1 - cos(a))

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.

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 . 

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 .

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.

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?

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.

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?

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 ?

Frame of escaping traveler and frame of arriving the same traveler have slower time relative each other than relative to home frame. Therefore traveled clock shows less time than home clock at meeting.