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2024-01-09 08:21:54 +0200 | commented answer | Large System of Quadratic Trigonometric Equations Yes it worked! Thanks a lot Max!! |

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2024-01-08 18:28:40 +0200 | commented answer | Large System of Quadratic Trigonometric Equations Yes indeed! Thanks for pointing that out. Luckily it did return the correct root but by chance I guess. Did not notice t |

2024-01-08 18:18:01 +0200 | commented answer | Large System of Quadratic Trigonometric Equations Yes indeed! Thanks for pointing that up. Luckily it did return the correct root but by chance I guess. Did not notice th |

2024-01-06 16:17:55 +0200 | commented answer | Large System of Quadratic Trigonometric Equations As an example take equation (1) and (2) from the original set of equations. Add (1) + l i * (2) yields: $$ 0 = l (4 e^ |

2024-01-05 16:31:39 +0200 | edited answer | Large System of Quadratic Trigonometric Equations I finally got to a result using a completely different approach. First I noticed $\rho$ and $\beta$ are one and the same |

2024-01-05 16:24:43 +0200 | edited answer | Large System of Quadratic Trigonometric Equations I finally got to a result using a completely different approach. First I noticed $\rho$ and $\beta$ are one and the same |

2024-01-05 16:21:23 +0200 | edited answer | Large System of Quadratic Trigonometric Equations I finally got to a result using a completely different approach. First I noticed $\rho$ and $\beta$ are one and the same |

2024-01-05 16:11:29 +0200 | marked best answer | Large System of Quadratic Trigonometric Equations Following the answers provided in 2 previous posts https://ask.sagemath.org/question/706... and https://ask.sagemath.org/question/726..., we tried to apply the proposed methods to a yet more complex system of trigonometric equations. We are now considering a system with one distance variable $l$, and 10 angle variables $\alpha$, $\beta$, $\gamma$, $\delta$, $\rho$, $a$, $b$, $c$, $d$ and $e$. The said trigonometric system of equations is consistuted of 11 equations as described below: $$ 0 = l \left( {4} \cos{\left({{\alpha}}\right)} + \cos{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} + \cos{\left({{\beta}}\right)} + \cos{\left({{\alpha} - \frac{{2} {\pi}}{{3}}}\right)} \right) - {1} $$ $$ 0 = {4} \sin{\left({{\alpha}}\right)} + \sin{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} + \sin{\left({{\beta}}\right)} + \sin{\left({{\alpha} - \frac{{2} {\pi}}{{3}}}\right)} $$ $$ 0 = {2} \cos{\left({{\alpha}}\right)} - \cos{\left({{\gamma}}\right)} - \cos{\left({{\delta}}\right)} - \cos{\left({{\rho}}\right)} + \cos{\left({{\alpha} + \frac{{2} {\pi}}{{3}}}\right)} $$ $$ 0 = l \left( {2} \sin{\left({{\alpha}}\right)} - \sin{\left({{\gamma}}\right)} - \sin{\left({{\delta}}\right)} - \sin{\left({{\rho}}\right)} + \sin{\left({{\alpha} + \frac{{2} {\pi}}{{3}}}\right)} \right) - {1} $$ $$ 0 = \cos{\left({a}\right)} - \cos{\left({b}\right)} - \cos{\left({c}\right)} + \cos{\left({{\gamma}}\right)} + \cos{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} $$ $$ 0 = \sin{\left({a}\right)} - \sin{\left({b}\right)} - \sin{\left({c}\right)} + \sin{\left({{\gamma}}\right)} + \sin{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} $$ $$ 0 = \cos{\left({c}\right)} + \cos{\left({e}\right)} - \cos{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} - \cos{\left({{\alpha}}\right)} + \cos{\left({{\delta}}\right)} $$ $$ 0 = \sin{\left({c}\right)} + \sin{\left({e}\right)} - \sin{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} - \sin{\left({{\alpha}}\right)} + \sin{\left({{\delta}}\right)} $$ $$ 0 = \cos{\left({b}\right)} + \cos{\left({d}\right)} - \cos{\left({{\rho}}\right)} - \cos{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} - \cos{\left({e}\right)} $$ $$ 0 = \sin{\left({b}\right)} + \sin{\left({d}\right)} - \sin{\left({{\rho}}\right)} - \sin{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} - \sin{\left({e}\right)} $$ $$ 0 = \cos{\left({{\rho}}\right)} + \cos{\left({{\alpha} - \frac{{\pi}}{{3}}}\right)} + \cos{\left({{\alpha}}\right)} - \cos{\left({{\gamma}}\right)} - \cos{\left({{\delta}}\right)} - \cos{\left({d}\right)} - \cos{\left({a}\right)} $$ as usual, we are interested in finding the minimal polynomial of the root $l$ between 0 and 1 for which a numerical appriximation is $l \approx 0.2226926944766917$. Below is my (unsuccessfull) attempt to use to solve this problem: Permalink for testing purposes: https://sagecell.sagemath.org/?z=eJx9... Additionnal Permalink using Elimination_ideal, also fails: https://sagecell.sagemath.org/?z=eJy1... In this case sagemathcell websocket kernell crashes at the following line In advance, thanks for your help. kr, |

2024-01-05 16:10:05 +0200 | edited answer | Large System of Quadratic Trigonometric Equations |

2024-01-05 16:07:36 +0200 | edited answer | Large System of Quadratic Trigonometric Equations |

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2024-01-05 14:47:13 +0200 | answered a question | Large System of Quadratic Trigonometric Equations |

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2023-11-12 12:07:35 +0200 | commented question | Large System of Quadratic Trigonometric Equations Thanks, I just tried but for this particular problem does not seem to be sufficient. Thanks though for the tip... |

2023-11-11 17:16:18 +0200 | commented question | Large System of Quadratic Trigonometric Equations @Max Alkseyev, I removed the dimension call and confirm that RR.ideal returns successfully. However, elimination_ideal d |

2023-11-11 17:14:40 +0200 | edited question | Large System of Quadratic Trigonometric Equations Large System of Quadratic Trigonometric Equations Following the answers provided in 2 previous posts https://ask.sagemat |

2023-11-07 16:57:02 +0200 | commented question | Large System of Quadratic Trigonometric Equations Thanks a lot! I will have a look at it. |

2023-11-07 15:24:13 +0200 | commented question | Large System of Quadratic Trigonometric Equations Wow !! That is remarkable ... Thank you so much for your help. Thanks for the explanations. Hopefully 7 days will be eno |

2023-11-07 14:50:42 +0200 | commented question | Large System of Quadratic Trigonometric Equations I don't understand. There is a print statement right after that print("Solution dimension = ", J1.dimension()), however |

2023-11-07 14:44:16 +0200 | commented question | Large System of Quadratic Trigonometric Equations I don't understand. There is a print statement right after that print("Solution dimension = ", J1.dimension()), however |

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2023-11-06 19:36:40 +0200 | edited question | Large System of Quadratic Trigonometric Equations Large System of Trigonometric Equations Following the answers provided in 2 previous posts https://ask.sagemath.org/ques |

2023-11-06 16:46:03 +0200 | marked best answer | Trigonometric System of Quadratic Equations Given the system of equations below with variables x, y, beta and r, my goal is to eliminate x, y and beta in order to have a polynomial equation which admits r as one of its roots. $$4r^2 = \left(\cos(\beta)+2\cos\left(-\frac{7\pi}{12}\right)\right)^2 r^2+\left(\frac{1}{2}+\left(\sin(\beta)+2\sin\left(\frac{-7\pi}{12}\right)\right)r\right)^2$$ $$ 4r^2 = \left(\frac{1}{2}-2\cos\left(\frac{-7\pi}{12}\right)r-x\right)^2+\left(-2\sin\left(\frac{-7\pi}{12}\right)r-y\right)^2$$ $$4r^2 = \left(\frac{1}{2}-\cos(\beta)r-2\cos\left(\frac{5\pi}{12}\right)r-x\right)^2+\left(\frac{1}{2}-\sin(\beta)r-2\sin\left(\frac{5\pi}{12}\right)r-y\right)^2$$ $$16r^2 = \left(x+1/2-2\cos\left(\frac{\pi}{12}\right)r\right)^2+\left(y-2\sin\left(\frac{\pi}{12}\right)r\right)^2$$ I tried to successively use solve in sage: Sadly however I am left with two equations in r and beta. I cannot seem to find a way to perform the final step of eliminating beta. I suppose this is due to the fact that beta appears as an argument of trigonometric functions. $$4 r^{2} = \frac{1}{16} {\left(2 r {\left(\sqrt{6} + \sqrt{2}\right)} - \frac{4 {\left(\sqrt{6} {\left(2 \sqrt{2}
\cos\left(\beta\right) - 1\right)} + 13 \, \sqrt{2} - 8 \cos\left(\beta\right) + 8 \sin\left(\beta\right)\right)} r^{3} - 4 {\left(\sqrt{6} {\left(\sqrt{2} - \cos\left(\beta\right) - \sin\left(\beta\right)\right)} + \sqrt{2} {\left(\cos\left(\beta\right)
+ \sin\left(\beta\right)\right)} + 2\right)} r^{2} - r {\left(\sqrt{6} - \sqrt{2}\right)}}{2 {\left(\sqrt{6} {\left(2 \sqrt{2} -
\cos\left(\beta\right) + \sin\left(\beta\right)\right)} + \sqrt{2} {\left(\cos\left(\beta\right) +
\sin\left(\beta\right)\right)}\right)} r^{2} - r {\left(\sqrt{6} + 3 \sqrt{2} + 2 \, \sin\left(\beta\right)\right)} + 1} -
2\right)}^{2} + \frac{1}{16} {\left(2 r {\left(\sqrt{6} - \sqrt{2}\right)} - \frac{4 {\left(\sqrt{6} {\left(2 \, \sqrt{2}
\sin\left(\beta\right) + 1\right)} + 13 \sqrt{2} - 8 \cos\left(\beta\right) + 8 \sin\left(\beta\right)\right)} r^{3} - 4 I know the approximate numerical solutions are as follows: x= 0.17943518672220243 y= 0.15109739303031233 beta= -0.42806907644466885 r= 0.11605914696138518 Any idea how to perform the last step, eliminate the variable beta and obtain a minimum polynomial for r? In advance thanks a lot for your Help! |

2023-11-06 16:46:02 +0200 | commented answer | Trigonometric System of Quadratic Equations Hi Ricardo, Truly sorry to solicit your attention again. I once again have a similar question but this time I believe m |

2023-11-05 19:23:03 +0200 | commented question | Large System of Quadratic Trigonometric Equations Permalink added in the Question Edit Box under the Sage Code Box |

2023-11-05 19:21:47 +0200 | commented question | Large System of Quadratic Trigonometric Equations Here is what I see in the browser network console Kernel: kernel_starting (null) 2kernel.js:107 Kernel: kernel_created |

2023-11-05 19:21:24 +0200 | commented question | Large System of Quadratic Trigonometric Equations Here is what I see in the browser network console Kernel: kernel_starting (null) 2kernel.js:107 Kernel: kernel_created |

2023-11-05 19:21:11 +0200 | edited question | Large System of Quadratic Trigonometric Equations Large System of Trigonometric Equations Following the answers provided in 2 previous posts https://ask.sagemath.org/ques |

2023-11-05 19:19:10 +0200 | commented question | Large System of Quadratic Trigonometric Equations Here is what I see in the browser network console Kernel: kernel_starting (null) 2kernel.js:107 Kernel: kernel_created |

2023-11-05 18:10:31 +0200 | commented question | Large System of Quadratic Trigonometric Equations Oh ok but, actually I do not have a local Sage installation. I am using the online version on https://sagecell.sagemath. |

2023-11-05 17:37:48 +0200 | edited question | Large System of Quadratic Trigonometric Equations Large System of Trigonometric Equations Following the answers provided in 2 previous posts https://ask.sagemath.org/ques |

2023-11-05 17:15:17 +0200 | edited question | Large System of Quadratic Trigonometric Equations |

2023-11-05 17:14:29 +0200 | edited question | Large System of Quadratic Trigonometric Equations |

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2023-11-05 17:05:40 +0200 | commented answer | Finding the minimal polynomial for a solution of a trigonometric system of quadratic equations Dear Mas, I now have a similar problem with this time some issue on the line J1 = RR.ideal(Sys2). I posted the descripti |

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