យើងមានតំលៃប្រហែល
c
o
s
(
π
7
)
≃
0
,
9009688...
≃
9
10
{\displaystyle cos({\frac {\pi }{7}})\simeq 0,9009688...\simeq {\frac {9}{10}}~}
តំលៃនេះអាចអោយយើងសង់បន្ទាត់ និង កំប៉ានៃមុំ ដែលមានរង្វាស់ជិតស្មើនឹង
π
7
{\displaystyle {\frac {\pi }{7}}}
យើងគូសអង្កត់
[
A
B
]
{\displaystyle \ [AB]}
A
P
=
9
10
A
B
{\displaystyle AP={\frac {9}{10}}AB~}
គេមានចំនុច C មួយជាចំនុចប្រសព្វរវាងរង្វង់ ផ្ចិត A កាំ AB ជាមួយនឹងបន្ទាត់កែង AB ត្រង់ P ។
ហេតុនេះមុំ
B
A
C
^
{\displaystyle {\widehat {BAC}}~}
π
7
{\displaystyle {\frac {\pi }{7}}}
X
3
−
X
2
−
2
X
+
1
=
0
{\displaystyle X^{3}-X^{2}-2X+1=0~}
មានឫស៖
{
2
cos
(
π
7
)
,
−
2
cos
(
2
π
7
)
,
2
cos
(
3
π
7
)
}
{\displaystyle \{2\cos({\frac {\pi }{7}}),-2\cos({\frac {2\pi }{7}}),2\cos({\frac {3\pi }{7}})\}~}
X
3
+
X
2
−
2
X
−
1
=
0
{\displaystyle X^{3}+X^{2}-2X-1=0~}
មានឫស៖
{
−
2
cos
(
π
7
)
,
2
cos
(
2
π
7
)
,
−
2
cos
(
3
π
7
)
}
{\displaystyle \{-2\cos({\frac {\pi }{7}}),2\cos({\frac {2\pi }{7}}),-2\cos({\frac {3\pi }{7}})\}~}
X
3
+
7
X
2
−
7
=
0
{\displaystyle X^{3}+{\sqrt {7}}X^{2}-{\sqrt {7}}=0~}
មានឫស៖
{
2
sin
(
π
7
)
,
−
2
sin
(
2
π
7
)
,
−
2
sin
(
3
π
7
)
}
{\displaystyle \{2\sin({\frac {\pi }{7}}),-2\sin({\frac {2\pi }{7}}),-2\sin({\frac {3\pi }{7}})\}~}
X
3
−
7
X
2
+
7
=
0
{\displaystyle X^{3}-{\sqrt {7}}X^{2}+{\sqrt {7}}=0~}
a pour racines :
{
−
2
sin
(
π
7
)
,
2
sin
(
2
π
7
)
,
2
sin
(
3
π
7
)
}
{\displaystyle \{-2\sin({\frac {\pi }{7}}),2\sin({\frac {2\pi }{7}}),2\sin({\frac {3\pi }{7}})\}~}
X
3
+
7
X
2
−
7
X
+
7
=
0
{\displaystyle X^{3}+{\sqrt {7}}X^{2}-7X+{\sqrt {7}}=0~}
មានឫស៖
{
tan
(
π
7
)
,
tan
(
2
π
7
)
,
−
tan
(
3
π
7
)
}
{\displaystyle \{\tan({\frac {\pi }{7}}),\tan({\frac {2\pi }{7}}),-\tan({\frac {3\pi }{7}})\}~}
X
3
−
7
X
2
−
7
X
−
7
=
0
{\displaystyle X^{3}-{\sqrt {7}}X^{2}-7X-{\sqrt {7}}=0~}
មានឫស៖
{
−
tan
(
π
7
)
,
−
tan
(
2
π
7
)
,
tan
(
3
π
7
)
}
{\displaystyle \{-\tan({\frac {\pi }{7}}),-\tan({\frac {2\pi }{7}}),\tan({\frac {3\pi }{7}})\}~}
cos
(
π
7
)
−
cos
(
2
π
7
)
+
cos
(
3
π
7
)
=
1
2
{\displaystyle \cos({\frac {\pi }{7}})-\cos({\frac {2\pi }{7}})+\cos({\frac {3\pi }{7}})={\frac {1}{2}}~}
cos
(
π
7
)
.
cos
(
2
π
7
)
.
cos
(
3
π
7
)
=
−
1
8
{\displaystyle \cos({\frac {\pi }{7}}).\cos({\frac {2\pi }{7}}).\cos({\frac {3\pi }{7}})=-{\frac {1}{8}}~}
cos
(
π
7
)
.
cos
(
2
π
7
)
−
cos
(
π
7
)
.
cos
(
3
π
7
)
+
cos
(
2
π
7
)
.
cos
(
3
π
7
)
=
−
1
2
{\displaystyle \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
(
π
7
)
−
sin
(
2
π
7
)
−
sin
(
3
π
7
)
=
−
7
2
{\displaystyle \sin({\frac {\pi }{7}})-\sin({\frac {2\pi }{7}})-\sin({\frac {3\pi }{7}})=-{\frac {\sqrt {7}}{2}}~}
sin
(
π
7
)
.
sin
(
2
π
7
)
.
sin
(
3
π
7
)
=
7
8
{\displaystyle \sin({\frac {\pi }{7}}).\sin({\frac {2\pi }{7}}).\sin({\frac {3\pi }{7}})={\frac {\sqrt {7}}{8}}~}
sin
(
π
7
)
.
sin
(
2
π
7
)
+
sin
(
π
7
)
.
sin
(
3
π
7
)
−
sin
(
2
π
7
)
.
sin
(
3
π
7
)
=
0
{\displaystyle \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
(
π
7
)
+
tan
(
2
π
7
)
−
tan
(
3
π
7
)
=
−
7
{\displaystyle \tan({\frac {\pi }{7}})+\tan({\frac {2\pi }{7}})-\tan({\frac {3\pi }{7}})=-{\sqrt {7}}~}
tan
(
π
7
)
.
tan
(
2
π
7
)
.
tan
(
3
π
7
)
=
7
{\displaystyle \tan({\frac {\pi }{7}}).\tan({\frac {2\pi }{7}}).\tan({\frac {3\pi }{7}})={\sqrt {7}}~}
tan
(
π
7
)
.
tan
(
2
π
7
)
−
tan
(
π
7
)
.
tan
(
3
π
7
)
−
tan
(
2
π
7
)
.
tan
(
3
π
7
)
=
−
7
{\displaystyle \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~}
cos
(
π
7
)
cos
(
2
π
7
)
=
1
2
cos
(
2
π
7
)
+
1
4
{\displaystyle \cos({\frac {\pi }{7}})\cos({\frac {2\pi }{7}})={\frac {1}{2}}\cos({\frac {2\pi }{7}})+{\frac {1}{4}}~}
cos
(
π
7
)
cos
(
3
π
7
)
=
1
2
cos
(
π
7
)
−
1
4
{\displaystyle \cos({\frac {\pi }{7}})\cos({\frac {3\pi }{7}})={\frac {1}{2}}\cos({\frac {\pi }{7}})-{\frac {1}{4}}~}
cos
(
2
π
7
)
cos
(
3
π
7
)
=
−
1
2
cos
(
3
π
7
)
+
1
4
{\displaystyle \cos({\frac {2\pi }{7}})\cos({\frac {3\pi }{7}})=-{\frac {1}{2}}\cos({\frac {3\pi }{7}})+{\frac {1}{4}}~}
cos
2
(
π
7
)
=
1
2
+
1
2
cos
(
2
π
7
)
{\displaystyle \cos ^{2}({\frac {\pi }{7}})={\frac {1}{2}}+{\frac {1}{2}}\cos({\frac {2\pi }{7}})~}
cos
2
(
2
π
7
)
=
1
2
−
1
2
cos
(
3
π
7
)
{\displaystyle \cos ^{2}({\frac {2\pi }{7}})={\frac {1}{2}}-{\frac {1}{2}}\cos({\frac {3\pi }{7}})~}
cos
2
(
3
π
7
)
=
1
2
−
1
2
cos
(
π
7
)
{\displaystyle \cos ^{2}({\frac {3\pi }{7}})={\frac {1}{2}}-{\frac {1}{2}}\cos({\frac {\pi }{7}})~}
ចំពោះតំលៃផ្សេងៗនៃ k ក្នុង kπ/7 គេអាចត្រលប់ទៅរូបមន្តមុន
cos
(
4
π
7
)
=
−
cos
(
3
π
7
)
{\displaystyle \cos({\frac {4\pi }{7}})=-\cos({\frac {3\pi }{7}})~}
cos
(
5
π
7
)
=
−
cos
(
2
π
7
)
{\displaystyle \cos({\frac {5\pi }{7}})=-\cos({\frac {2\pi }{7}})~}
cos
(
6
π
7
)
=
−
cos
(
π
7
)
{\displaystyle \cos({\frac {6\pi }{7}})=-\cos({\frac {\pi }{7}})~}
sin
(
4
π
7
)
=
sin
(
3
π
7
)
{\displaystyle \sin({\frac {4\pi }{7}})=\sin({\frac {3\pi }{7}})~}
sin
(
5
π
7
)
=
sin
(
2
π
7
)
{\displaystyle \sin({\frac {5\pi }{7}})=\sin({\frac {2\pi }{7}})~}
sin
(
6
π
7
)
=
sin
(
π
7
)
{\displaystyle \sin({\frac {6\pi }{7}})=\sin({\frac {\pi }{7}})~}
tan
(
4
π
7
)
=
−
tan
(
3
π
7
)
{\displaystyle \tan({\frac {4\pi }{7}})=-\tan({\frac {3\pi }{7}})~}
tan
(
5
π
7
)
=
−
tan
(
2
π
7
)
{\displaystyle \tan({\frac {5\pi }{7}})=-\tan({\frac {2\pi }{7}})~}
tan
(
6
π
7
)
=
−
tan
(
π
7
)
{\displaystyle \tan({\frac {6\pi }{7}})=-\tan({\frac {\pi }{7}})~}
យើងមាន
∀
k
∈
Z
,
2
k
(
cos
k
(
π
7
)
+
cos
k
(
3
π
7
)
+
cos
k
(
5
π
7
)
)
∈
N
∗
{\displaystyle \forall k\in \mathbb {Z} ,\quad 2^{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
(
cos
(
π
7
)
+
cos
(
3
π
7
)
+
cos
(
5
π
7
)
)
=
1
{\displaystyle 2\left(\cos({\frac {\pi }{7}})+\cos({\frac {3\pi }{7}})+\cos({\frac {5\pi }{7}})\right)=1~}
2
2
(
cos
2
(
π
7
)
+
cos
2
(
3
π
7
)
+
cos
2
(
5
π
7
)
)
=
5
{\displaystyle 2^{2}\left(\cos ^{2}({\frac {\pi }{7}})+\cos ^{2}({\frac {3\pi }{7}})+\cos ^{2}({\frac {5\pi }{7}})\right)=5~}
2
3
(
cos
3
(
π
7
)
+
cos
3
(
3
π
7
)
+
cos
3
(
5
π
7
)
)
=
4
{\displaystyle 2^{3}\left(\cos ^{3}({\frac {\pi }{7}})+\cos ^{3}({\frac {3\pi }{7}})+\cos ^{3}({\frac {5\pi }{7}})\right)=4~}
2
4
(
cos
4
(
π
7
)
+
cos
4
(
3
π
7
)
+
cos
4
(
5
π
7
)
)
=
13
{\displaystyle 2^{4}\left(\cos ^{4}({\frac {\pi }{7}})+\cos ^{4}({\frac {3\pi }{7}})+\cos ^{4}({\frac {5\pi }{7}})\right)=13~}
2
5
(
cos
5
(
π
7
)
+
cos
5
(
3
π
7
)
+
cos
5
(
5
π
7
)
)
=
16
{\displaystyle 2^{5}\left(\cos ^{5}({\frac {\pi }{7}})+\cos ^{5}({\frac {3\pi }{7}})+\cos ^{5}({\frac {5\pi }{7}})\right)=16~}
2
6
(
cos
6
(
π
7
)
+
cos
6
(
3
π
7
)
+
cos
6
(
5
π
7
)
)
=
38
{\displaystyle 2^{6}\left(\cos ^{6}({\frac {\pi }{7}})+\cos ^{6}({\frac {3\pi }{7}})+\cos ^{6}({\frac {5\pi }{7}})\right)=38~}
2
7
(
cos
7
(
π
7
)
+
cos
7
(
3
π
7
)
+
cos
7
(
5
π
7
)
)
=
57
{\displaystyle 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
1
2
(
1
cos
(
π
7
)
+
1
cos
(
3
π
7
)
+
1
cos
(
5
π
7
)
)
=
2
{\displaystyle {\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~}
1
2
2
(
1
cos
2
(
π
7
)
+
1
cos
2
(
3
π
7
)
+
1
cos
2
(
5
π
7
)
)
=
6
{\displaystyle {\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~}
1
2
3
(
1
cos
3
(
π
7
)
+
1
cos
3
(
3
π
7
)
+
1
cos
3
(
5
π
7
)
)
=
11
{\displaystyle {\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~}
1
2
4
(
1
cos
4
(
π
7
)
+
1
cos
4
(
3
π
7
)
+
1
cos
4
(
5
π
7
)
)
=
26
{\displaystyle {\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~}
1
2
5
(
1
cos
5
(
π
7
)
+
1
cos
5
(
3
π
7
)
+
1
cos
5
(
5
π
7
)
)
=
57
{\displaystyle {\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។
![{\displaystyle \ [AB]}](/media/api/rest_v1/media/math/render/svg/e792dbd456399f4cc87cb68feb7fef880ca06a95)
