DimaMazin Posted April 18, 2023 Share Posted April 18, 2023 a = (Pi - 2)/4 + sin(a) a is unknown angle(rad) in unit circle a/2 is area of sector of angle a (Pi-2)/8 is area of segment of angle a Link to comment Share on other sites More sharing options...
joigus Posted April 18, 2023 Share Posted April 18, 2023 This is a transcendental equation. You cannot solve but by means of approximate methods, like iterations that are known to converge to a solution, etc. Link to comment Share on other sites More sharing options...
John Cuthber Posted April 18, 2023 Share Posted April 18, 2023 I wonder if there's a geometrical solution. Link to comment Share on other sites More sharing options...
studiot Posted April 18, 2023 Share Posted April 18, 2023 (edited) Firstly @DimaMazin Please number your equations for discussion in future. Calling them 1, 2 and 3 in the order in which they appear in the OP, This is not a valid question. Equation 1 may be simply derived from equation 3 and is an identity so valid for all values of a, where a is measured in radians. Since I can no longer use either of my maths computers here you will all have to put up with crappy english language. The standard expression for the area, A, of a segment of a circle of radius R is A = R^2/2 (a - sina) Since you say R = 1 We have (Pi - 2) / 8 = 1/2 (a - sina) Multiply through by 2 ( Pi - 2 ) / 4 = a - sina Which is your equation 1 Perhaps you are looking for an angle a where the area is the same for both sector and segment ? Edited April 18, 2023 by studiot Link to comment Share on other sites More sharing options...
Genady Posted April 18, 2023 Share Posted April 18, 2023 a=1.226819... Link to comment Share on other sites More sharing options...
studiot Posted April 18, 2023 Share Posted April 18, 2023 33 minutes ago, Genady said: a=1.226819... How does that work out with Dima's equation2, given that the standard area for a sector is Ra ? Link to comment Share on other sites More sharing options...
Genady Posted April 18, 2023 Share Posted April 18, 2023 5 minutes ago, studiot said: How does that work out with Dima's equation2, given that the standard area for a sector is Ra ? Area of a sector with angle a in radians and unit radius is a/(2p)*(p*r^2)=a/2 for any a. 2 Link to comment Share on other sites More sharing options...
studiot Posted April 18, 2023 Share Posted April 18, 2023 37 minutes ago, Genady said: Area of a sector with angle a in radians and unit radius is a/(2p)*(p*r^2)=a/2 for any a. Yes you are right, the formula is R^2a/2 = a/2 in this case, so it does fit +1 Link to comment Share on other sites More sharing options...
joigus Posted April 19, 2023 Share Posted April 19, 2023 Genady is right. It's not a transcendental equation. I made a mistake. Sorry. Somehow I thought I saw an "a" in the first term, which wasn't there. Link to comment Share on other sites More sharing options...
Genady Posted April 19, 2023 Share Posted April 19, 2023 5 hours ago, joigus said: Genady is right. It's not a transcendental equation. I made a mistake. Sorry. Somehow I thought I saw an "a" in the first term, which wasn't there. I think, you were right, and it is a transcendental equation. I've solved it with a very simple approximation procedure and Excel. Here it is: f(x) = (Pi - 2)/4 + sin(x) - x 2 Link to comment Share on other sites More sharing options...
joigus Posted April 19, 2023 Share Posted April 19, 2023 4 hours ago, Genady said: I think, you were right, and it is a transcendental equation. I've solved it with a very simple approximation procedure and Excel. Here it is: Yes, thank you. It is. Every time I look at this thread I look without looking, if you know what I mean. a = const. + sin(a) is a transcendental equation. Doh!! Link to comment Share on other sites More sharing options...
DimaMazin Posted April 21, 2023 Author Share Posted April 21, 2023 I have got so complex equation for definition of sine of the angle a ,but I am not sure it is correct. 4sin2 +4sin*cos-2sin-Pi*sin+Pi*cos-2cos=0 If we use Genady's definition then we can approximately check it. I have used method of disproportionate division of segment of angle Pi/2 (area of wich is (Pi-2)/4) and second part of the sector of the angle Pi/2 ,area of wich is 1/2. It is when angle a and angle Pi/2 -a disproportionately divide the parts. Then every of the 4 parts has the same disproportionate unknown(for 2 parts it is u and for 2 other parts it is -u). It is not working for definition sine and cosine of 1 radian relative to Pi, therefore rather it is nonsense. Link to comment Share on other sites More sharing options...
DimaMazin Posted June 12, 2023 Author Share Posted June 12, 2023 (edited) On 4/18/2023 at 11:24 AM, John Cuthber said: I wonder if there's a geometrical solution. Yes. But we can solve simpler problem. a=2sin(a) ((sin-P/4)*2-sin*(-cos))*2 = sin-2sin*(-cos)-4(sin-P/4) sin = (2/7)P a=(4/7)P Area of segment of angle a = (1/7)P Area of segment of angle (3/14)P = (1/14)P Exuse me. I have mistaken again. I incorectly made the equation. In correct equation the variables annihilate. Edited June 12, 2023 by DimaMazin Link to comment Share on other sites More sharing options...
DimaMazin Posted June 22, 2023 Author Share Posted June 22, 2023 I think the simplest problem, of the similars, is sector area = chord2 a/2 = 2-2cos(a) But I don't know how to solve it. Link to comment Share on other sites More sharing options...
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