DimaMazin

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?

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 .

Seems that is wrong. I should remake it.

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.

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))] 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)))]

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))]

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))

Yes and proportional divisions can be any , but coordinates of cross point should be constant.

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 .

Arc is angle(rad). I made some explore. If it exists then rather it is between 5/6 Pi and 17/18 Pi .

I should not louse time. I should define arc of definition.

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?

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.

What is exact sin(Pi/4)=21/2/2 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 (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 ?

Because it has different density or properties.