រូបមន្តត្រីកោណមាត្រត្រង់ kπ/7

## តំលៃប្រហែល 
យើងមានតំលៃប្រហែល
: cos(\frac{\pi}{7}) \simeq 0,9009688... \simeq \frac{9}{10} ~
តំលៃនេះអាចអោយយើងសង់បន្ទាត់ និង កំប៉ានៃមុំ ដែលមានរង្វាស់ជិតស្មើនឹង   \frac{\pi}{7}  ។

ស្តាំ
យើងគូសអង្កត់ \ [AB]   និង ចំនុច P មួយដែល
: AP = \frac{9}{10}AB ~

គេមានចំនុច C មួយជាចំនុចប្រសព្វរវាងរង្វង់​ផ្ចិត A កាំ AB ជាមួយនឹងបន្ទាត់កែង AB ត្រង់ P ។

ហេតុនេះមុំ   \widehat{BAC} ~ មានរង្វាស់ប្រហែលនឹង  \frac{\pi}{7}  ។

## ដំណោះស្រាយសមីការមួយចំនួន 
* សមីការ​៖

 X^3 - X^2 -2X +1 = 0 ~

មានឫស៖

 \{2\cos(\frac{\pi}{7}), -2\cos(\frac{2\pi}{7}), 2\cos(\frac{3\pi}{7})\} ~

* សមីការ៖

 X^3 + X^2 -2X - 1 = 0 ~

មានឫស៖

 \{-2\cos(\frac{\pi}{7}),2\cos(\frac{2\pi}{7}),-2\cos(\frac{3\pi}{7})\} ~

* សមីការ៖

 X^3 + \sqrt{7}X^2 - \sqrt{7} = 0 ~

មានឫស៖
 \{2\sin(\frac{\pi}{7}),-2\sin(\frac{2\pi}{7}),-2\sin(\frac{3\pi}{7})\} ~

* សមីការ៖

 X^3 - \sqrt{7}X^2 + \sqrt{7} = 0 ~

a pour racines :

 \{-2\sin(\frac{\pi}{7}),2\sin(\frac{2\pi}{7}),2\sin(\frac{3\pi}{7})\} ~

* សមីការ៖

 X^3 + \sqrt{7}X^2 - 7X + \sqrt{7} = 0 ~

មានឫស៖

 \{\tan(\frac{\pi}{7}),\tan(\frac{2\pi}{7}),-\tan(\frac{3\pi}{7})\} ~

* សមីការ៖

 X^3 - \sqrt{7}X^2 - 7X - \sqrt{7} = 0 ~

មានឫស៖

 \{-\tan(\frac{\pi}{7}),-\tan(\frac{2\pi}{7}),\tan(\frac{3\pi}{7})\} ~

## រូបមន្តអូម៉ូសែន 
 \cos(\frac{\pi}{7}) - \cos(\frac{2\pi}{7}) + \cos(\frac{3\pi}{7}) = \frac{1}{2}~

 \cos(\frac{\pi}{7}) . \cos(\frac{2\pi}{7}) . \cos(\frac{3\pi}{7}) = -\frac{1}{8}~

 \cos(\frac{\pi}{7}).\cos(\frac{2\pi}{7})  - \cos(\frac{\pi}{7}).\cos(\frac{3\pi}{7}) + \cos(\frac{2\pi}{7}).\cos(\frac{3\pi}{7}) = -\frac{1}{2}~

 \sin(\frac{\pi}{7}) - \sin(\frac{2\pi}{7}) - \sin(\frac{3\pi}{7}) = -\frac{\sqrt{7}}{2}~

 \sin(\frac{\pi}{7}) . \sin(\frac{2\pi}{7}) . \sin(\frac{3\pi}{7}) = \frac{\sqrt{7}}{8}~

 \sin(\frac{\pi}{7}).\sin(\frac{2\pi}{7})  + \sin(\frac{\pi}{7}).\sin(\frac{3\pi}{7}) - \sin(\frac{2\pi}{7}).\sin(\frac{3\pi}{7}) = 0 ~

 \tan(\frac{\pi}{7}) + \tan(\frac{2\pi}{7}) - \tan(\frac{3\pi}{7}) = -\sqrt{7}~

 \tan(\frac{\pi}{7}) . \tan(\frac{2\pi}{7}) . \tan(\frac{3\pi}{7}) = \sqrt{7}~

 \tan(\frac{\pi}{7}).\tan(\frac{2\pi}{7})  - \tan(\frac{\pi}{7}).\tan(\frac{3\pi}{7}) - \tan(\frac{2\pi}{7}).\tan(\frac{3\pi}{7}) = -7~

## រូបមន្តលីនែអ៊ែរ 
 \cos(\frac{\pi}{7})\cos(\frac{2\pi}{7}) = \frac{1}{2}\cos(\frac{2\pi}{7}) + \frac{1}{4} ~

 \cos(\frac{\pi}{7})\cos(\frac{3\pi}{7}) = \frac{1}{2}\cos(\frac{\pi}{7}) - \frac{1}{4} ~

 \cos(\frac{2\pi}{7})\cos(\frac{3\pi}{7}) = -\frac{1}{2}\cos(\frac{3\pi}{7}) + \frac{1}{4} ~

 \cos^2(\frac{\pi}{7}) =\frac{1}{2} + \frac{1}{2}\cos(\frac{2\pi}{7}) ~

 \cos^2(\frac{2\pi}{7}) =\frac{1}{2} - \frac{1}{2}\cos(\frac{3\pi}{7}) ~

 \cos^2(\frac{3\pi}{7}) =\frac{1}{2} - \frac{1}{2}\cos(\frac{\pi}{7}) ~

## រូបមន្តបន្ថយ 
ចំពោះតំលៃផ្សេងៗនៃ k ក្នុង kπ/7 គេអាចត្រលប់ទៅរូបមន្តមុន

 \cos(\frac{4\pi}{7}) = -\cos(\frac{3\pi}{7}) ~

 \cos(\frac{5\pi}{7}) = -\cos(\frac{2\pi}{7}) ~
 
 \cos(\frac{6\pi}{7}) = -\cos(\frac{\pi}{7}) ~

 \sin(\frac{4\pi}{7}) = \sin(\frac{3\pi}{7}) ~

 \sin(\frac{5\pi}{7}) = \sin(\frac{2\pi}{7}) ~
 
 \sin(\frac{6\pi}{7}) = \sin(\frac{\pi}{7}) ~

 \tan(\frac{4\pi}{7}) = -\tan(\frac{3\pi}{7}) ~

 \tan(\frac{5\pi}{7}) = -\tan(\frac{2\pi}{7}) ~
 
 \tan(\frac{6\pi}{7}) = -\tan(\frac{\pi}{7}) ~

## លក្ខណៈគួរកត់សំគាល់ 
យើងមាន

 \forall k \in \mathbb{Z}, \quad2^k\left(\cos^k(\frac{\pi}{7}) + \cos^k(\frac{3\pi}{7}) + \cos^k(\frac{5\pi}{7}) \right) \in \mathbb{N}^* ~

* ចំពោះតំលៃដំបូងនៃ k វិជ្ជមាន គេទទួលបាន

 2\left(\cos(\frac{\pi}{7}) + \cos(\frac{3\pi}{7}) + \cos(\frac{5\pi}{7})\right) = 1 ~

 2^2\left(\cos^2(\frac{\pi}{7}) + \cos^2(\frac{3\pi}{7}) + \cos^2(\frac{5\pi}{7})\right) = 5 ~

 2^3\left(\cos^3(\frac{\pi}{7}) + \cos^3(\frac{3\pi}{7}) + \cos^3(\frac{5\pi}{7})\right) = 4 ~

 2^4\left(\cos^4(\frac{\pi}{7}) + \cos^4(\frac{3\pi}{7}) + \cos^4(\frac{5\pi}{7})\right) = 13 ~

 2^5\left(\cos^5(\frac{\pi}{7}) + \cos^5(\frac{3\pi}{7}) + \cos^5(\frac{5\pi}{7})\right) = 16 ~

 2^6\left(\cos^6(\frac{\pi}{7}) + \cos^6(\frac{3\pi}{7}) + \cos^6(\frac{5\pi}{7})\right) = 38 ~

 2^7\left(\cos^7(\frac{\pi}{7}) + \cos^7(\frac{3\pi}{7}) + \cos^7(\frac{5\pi}{7})\right) = 57 ~

។ល។

* ចំពោះតំលៃដំបូងនៃ k អវិជ្ជមាន គេទទួលបាន

 \frac{1}{2}\left(\frac{1}{\cos(\frac{\pi}{7})} + \frac{1}{\cos(\frac{3\pi}{7})} + \frac{1}{\cos(\frac{5\pi}{7})}\right) = 2 ~

 \frac{1}{2^2}\left(\frac{1}{\cos^2(\frac{\pi}{7})} + \frac{1}{\cos^2(\frac{3\pi}{7})} + \frac{1}{\cos^2(\frac{5\pi}{7})}\right) = 6 ~

 \frac{1}{2^3}\left(\frac{1}{\cos^3(\frac{\pi}{7})} + \frac{1}{\cos^3(\frac{3\pi}{7})} + \frac{1}{\cos^3(\frac{5\pi}{7})}\right) = 11 ~

 \frac{1}{2^4}\left(\frac{1}{\cos^4(\frac{\pi}{7})} + \frac{1}{\cos^4(\frac{3\pi}{7})} + \frac{1}{\cos^4(\frac{5\pi}{7})}\right) = 26 ~

 \frac{1}{2^5}\left(\frac{1}{\cos^5(\frac{\pi}{7})} + \frac{1}{\cos^5(\frac{3\pi}{7})} + \frac{1}{\cos^5(\frac{5\pi}{7})}\right) = 57 ~

។ល។