How to define arc of definition?

[studiot]

12 hours ago, DimaMazin said:

Draw unit circle x²+y²=1     Mark angle 166 degrees or 168 degrees .

 

Here is my attempt to follow your instructions.
I have used 168 degrees since this is actually divisible by 6.

The arc looks like semicircle, but is actually 168^o.

Is joining the points as instructed supposed to create straight lines that all meet at one point?
 

How does this help define a  sine ?

And more particularly the sine of what angle?

 

It is clear you have several members interested in a  technical discussion about this so it is in your own interests to engage as fully as you can.

 

[DimaMazin]

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))

 

 

Edited March 3, 2020 by DimaMazin

[studiot]

1 hour ago, DimaMazin said:

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))

 

 

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.

[DimaMazin]

 

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.

Edited March 3, 2020 by DimaMazin

[studiot]

On 2/29/2020 at 4:49 PM, DimaMazin said:

I didn't like approximate definitions of trigonometric functions (it was about 34 years ago).

Like most members here I understood you wish to discuss a non approximate definition of trigonometric functions.

I understood your 'arc of definition' to be an arc that somehow defines a trigonometric function.

Is this the case  or do you mean something else ?

Schoolboys are taught exact and perfect definitions of trig functions.
What is wrong with these?

Edited March 3, 2020 by studiot

[DimaMazin]

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))] 

Edited March 5, 2020 by DimaMazin

[DimaMazin]

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)))] 

[Ghideon]

On 3/3/2020 at 7:35 AM, DimaMazin said:

arc of definition

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

[DimaMazin]

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.

Edited March 5, 2020 by DimaMazin

[studiot]

51 minutes ago, Ghideon said:

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

 

22 minutes ago, DimaMazin said:

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.

 

So it seems that all you really want to do is present some trigonometry enabling someone to calculate an angle or its sine or cosine using a  complicated forumula.

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

 

One very big and fundamental difference between the standard method and yours is that yours cannot be used without a coordinate system.

Angles are and should be, independent of any coordinate system.
This is the way conventional definitions work.
That is it is a property of the standard method, using only triangles.

[taeto]

1 hour ago, DimaMazin said:

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.

But even without knowing the trigonometric functions and their inverses, is it not possible to still understand that an arc is a connected subset of points of the unit circle?

[Ghideon]

1 hour ago, DimaMazin said:

Yes

Thanks for confirming.

1 hour ago, DimaMazin said:

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.

Probably a language issue then, I know and understand a few things about arcs, circles and trigonometry. But I have been unable to find anything about "Arc of definition" in math. 

[DimaMazin]

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.

[taeto]

45 minutes ago, DimaMazin said:

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.

I do not understand why you say that an arc is "very complex". If we already agree about what is the unit circle, what is difficult about saying that an arc is a certain piece of the circle?

Are you aware of the difference between an "arc" and an "angle"?

[studiot]

1 hour ago, DimaMazin said:

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.

 

26 minutes ago, taeto said:

I do not understand why you say that an arc is "very complex". If we already agree about what is the unit circle, what is difficult about saying that an arc is a certain piece of the circle?

Are you aware of the difference between an "arc" and an "angle"?

 

Are you referring to this?

 

Quote

The unit circle

Since in any circle the same ratio of arc to radius determines a unique central angle, then for theoretical work we often use the unit circle, which is a circle of radius 1:  r = 1.

In the unit circle, the radian measure is the length of the arc s.  The length of that arc is a real number x.

s = rθ = 1· x = x.

We can identify radian measure, then, as the length x of an arc of the unit circle.  And it is here that the term trigonometric "function" has its full meaning. For, corresponding to each real number x -- each radian measure, each arc -- there is a unique value of sin x, of cos x, and so on. The definition of a function is satisfied. (Topic 3 of Precalculus.)

Moreover, when we draw the graph of y = sin x (Topic 18), we can imagine the unit circle rolled out in both directions onto the x-axis, and in that way marking the coördinates π, 2π;, −π, −2π, and so on, on the x-axis.

Because radian measure can be identified as an arc, the inverse trigonometric functions have their names. "arcsin" is the arc -- the radian measure -- whose sine is a certain number.

The ratio

sin x   x

In the unit circle, the opposite side AB is sin x.

sin x

=

AB  1

=  AB.

One of the main theorems in calculus concerns the ratio 

sin x    x

for very small values of x.  And we can see that when the point A on the circumference is very close to C -- that is, when the central angle AOC is very, very small -- then the opposite side AB will be virtually indistinguishable from the arc length AC.  That is,

sin x		x
sin x    x		1.

http://www.themathpage.com/aTrig/arc-length.htm#arc

[taeto]

12 minutes ago, studiot said:

Are you referring to this? http://www.themathpage.com/aTrig/arc-length.htm#arc

Not really. If we want to be painfully clear, then we should distinguish between what is an "arc" and what is an "angle", and your site does not seem willing to do just that.

The page certainly does not like to consider that an arc is just a bunch, each with a particular property, of points taken from the circle, which is the view that I intended. 

Angles define some special kinds of arcs. If you have two straight lines through the origin, you have angles that partition the circle into arcs, usually four of them. Each angle defines a separate arc. There is no involvement whatsoever of real numbers yet. So you should explain the need to introduce real numbers in the first place. They seem largely irrelevant to the introduction of these terms.    

[studiot]

53 minutes ago, taeto said:

The page certainly does not like to consider that an arc is just a bunch, each with a particular property, of points taken from the circle, which is the view that I intended.

You have to be careful defining an arc in that way in the case where you have a complete circle, you have one too many 'points'.

In any event my post was really aimed at Dima.

I was still trying to understand what he wants to do.

I included you because I hoped you would chip in, you are usually so helpful.

I did not want to start an argument.

 

My simple definition of an arc is a segment of a non self-intersecting curve, as opposed to a line segment being a part of a straight line.
I was not proposing to go into the niceties of Jordan and other curves.

This definition of an angle as the ratio of two lengths is the reason often given as to why an angle posesses no physical units (dimensions).

So angle = ArcLength/radius.

Edited March 5, 2020 by studiot

[taeto]

2 minutes ago, studiot said:

You have to be careful defining an arc in that way in the case where you have a complete circle, you have one too many 'points'.

In any event my post was really aimed at Dima.

I was still trying to understand what he wants to do.

I included you because I hoped you would chip in, you are usually so helpful.

I did not want to start an argument.

 

My simple definition of an arc is a segment of a non self-intersecting curve, as opposed to a line segment being a part of a straight line.
I was not proposing to go into the niceties of Jordan and other curves.

Sure, segment is fine. I do not want to start an argument either. Just make the point clear whether we are going into real numbers or not. In basic geometry it takes a few steps before you get to Archimedean and even Cantor-Dedekind properties. It seems to me that Dima wants to discuss at the level of measures of angles and real-valued trigonometric functions, which is many levels away from talking about arcs and angles. 

 

[DimaMazin]

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.

[DimaMazin]

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 .

[Bufofrog]

6 hours ago, DimaMazin said:

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

It won't.  There is no exact value for Pi.

[Strange]

11 minutes ago, Bufofrog said:

It won't.  There is no exact value for Pi.

Well, there is an exact value. You just can't write it down as a number.

[Ghideon]

6 hours ago, DimaMazin said:

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

Can you explain what that means? If your definition is identical to Pi then it is some redundant definition. If you get another value in your definition of Pi then your definition is wrong; it is not Pi anymore.

As I said in a post above: pi, defined as circumference/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. 

If you need to write Pi as a number maybe you could have look at base-Pi numbers at wikipedia.org/Non-integer_representation#Base_π or the source at http://datagenetics.com/blog/december22015/index.html

Quote

Base π can be used to more easily show the relationship between the diameter of a circle to its circumference, which corresponds to its perimeter; since circumference = diameter × π, a circle with a diameter 1_π will have a circumference of 10_π

 

Edited March 14, 2020 by Ghideon
spelling and quote

[studiot]

On 3/5/2020 at 3:31 PM, studiot said:

Are you referring to this?

@DimaMazin I am still waiting for an answer.

[taeto]

48 minutes ago, Ghideon said:

As I said in a post above: pi, defined as circumference/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. 

Every real number has a "non ending" representation in its decimal expansion. E.g. the standard representation of 1 is \(.999\ldots\) in base 10. You want to refer to the property that \(\pi\) has as opposed to rational numbers, that its representation is "non repeating".