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מתמטיקה תיכונית/טריגונומטריה/ערכים טריגונומטריים מדויקים
שפה
מעקב
עריכה
<
מתמטיקה תיכונית
|
טריגונומטריה
תוכן עניינים
1
0°:
2
1.5°: מצולע בעל 120 צלעות
3
1.875°: מצולע בעל 96 צלעות
4
2.25°: מצולע בעל 80 צלעות
5
2.8125°: מצולע בעל 64 צלעות
6
3°: מצולע בעל 60 צלעות
7
3.75°: מצולע בעל 48 צלעות
8
4.5°: מצולע בעל 40 צלעות
9
5.625°: מצולע בעל 32 צלעות
10
6°: מצולע בעל 30 צלעות
11
7.5°: מצולע בעל 24 צלעות
12
9°: מצולע בעל 20 צלעות
13
11.25°: מצולע בעל 16 צלעות
14
12°: מצולע בעל 15 צלעות
15
15°: מתורסר, מצולע בעל 12 צלעות
16
18°: מעושר, מצולע בעל 10 צלעות
17
סכום זויות: 21° = 9° + 12°
18
22.5°: מתומן
19
סכום זויות: 24° = 12° + 12°
20
סכום זויות: 27° = 12° + 15°
21
30°: משושה
22
סכום זויות: 33° = 15° + 18°
23
36°: מחומש
24
סכום זויות: 39° = 18° + 21°
25
סכום זויות: 42° = 21° + 21°
26
45°: ריבוע
27
סכום זויות: 54° = 27° + 27°
28
60°: משולש שוה-צלעות
29
סכום זויות: 67.5° = 7.5° + 60°
30
סכום זויות: 72° = 36° + 36°
31
סכום זויות: 75° = 30° + 45°
32
90°: זוית ישרה
0°:
עריכה
sin
(
0
∘
)
=
0
{\displaystyle \sin(0^{\circ })=0}
cos
(
0
∘
)
=
1
{\displaystyle \cos(0^{\circ })=1}
tan
(
0
∘
)
=
0
{\displaystyle \tan(0^{\circ })=0}
cot
(
0
∘
)
=
undefined
{\displaystyle \cot(0^{\circ })={\text{ undefined}}}
1.5°: מצולע בעל 120 צלעות
עריכה
sin
(
π
120
)
=
sin
(
1.5
∘
)
=
2
+
2
(
15
+
3
−
10
−
2
5
)
−
2
−
2
(
30
−
6
5
+
5
+
1
)
16
{\displaystyle \sin \left({\frac {\pi }{120}}\right)=\sin(1.5^{\circ })={\frac {{\sqrt {2+{\sqrt {2}}}}\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)-{\sqrt {2-{\sqrt {2}}}}\left({\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1\right)}{16}}}
cos
(
π
120
)
=
cos
(
1.5
∘
)
=
2
+
2
(
30
−
6
5
+
5
+
1
)
+
2
−
2
(
15
+
3
−
10
−
2
5
)
16
{\displaystyle \cos \left({\frac {\pi }{120}}\right)=\cos(1.5^{\circ })={\frac {{\sqrt {2+{\sqrt {2}}}}\left({\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1\right)+{\sqrt {2-{\sqrt {2}}}}\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)}{16}}}
1.875°: מצולע בעל 96 צלעות
עריכה
sin
(
π
96
)
=
sin
(
1.875
∘
)
=
2
−
2
+
2
+
2
+
3
2
{\displaystyle \sin \left({\frac {\pi }{96}}\right)=\sin(1.875^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}{2}}}
cos
(
π
96
)
=
cos
(
1.875
∘
)
=
2
+
2
+
2
+
2
+
3
2
{\displaystyle \cos \left({\frac {\pi }{96}}\right)=\cos(1.875^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}{2}}}
2.25°: מצולע בעל 80 צלעות
עריכה
sin
(
π
80
)
=
sin
(
2.25
∘
)
=
(
2
+
2
−
1
)
2
(
5
−
5
)
(
2
+
2
+
2
)
−
(
5
+
1
)
(
2
+
2
+
1
)
2
−
2
+
2
8
{\displaystyle \sin \left({\frac {\pi }{80}}\right)=\sin(2.25^{\circ })={\frac {{\Big (}{\sqrt {2+{\sqrt {2}}}}-1{\Big )}{\sqrt {2{\big (}5-{\sqrt {5}}{\big )}{\Big (}2+{\sqrt {2+{\sqrt {2}}}}{\Big )}}}-{\big (}{\sqrt {5}}+1{\big )}{\Big (}{\sqrt {2+{\sqrt {2}}}}+1{\Big )}{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}{8}}}
cos
(
π
80
)
=
cos
(
2.25
∘
)
=
8
+
2
8
+
2
8
+
2
10
+
2
5
4
{\displaystyle \cos \left({\frac {\pi }{80}}\right)=\cos(2.25^{\circ })={\frac {\sqrt {8+2{\sqrt {8+2{\sqrt {8+2{\sqrt {10+2{\sqrt {5}}}}}}}}}}{4}}}
2.8125°: מצולע בעל 64 צלעות
עריכה
sin
(
π
64
)
=
sin
(
2.8125
∘
)
=
2
−
2
+
2
+
2
+
2
2
{\displaystyle \sin \left({\frac {\pi }{64}}\right)=\sin(2.8125^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}{2}}}
cos
(
π
64
)
=
cos
(
2.8125
∘
)
=
2
+
2
+
2
+
2
+
2
2
{\displaystyle \cos \left({\frac {\pi }{64}}\right)=\cos(2.8125^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}{2}}}
3°: מצולע בעל 60 צלעות
עריכה
sin
(
π
60
)
=
sin
(
3
∘
)
=
2
(
1
−
3
)
5
+
5
+
2
(
5
−
1
)
(
3
+
1
)
16
{\displaystyle \sin \left({\frac {\pi }{60}}\right)=\sin(3^{\circ })={\frac {2{\big (}1-{\sqrt {3}}{\big )}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}{\big (}{\sqrt {5}}-1{\big )}{\big (}{\sqrt {3}}+1{\big )}}{16}}}
cos
(
π
60
)
=
cos
(
3
∘
)
=
2
(
3
+
1
)
5
+
5
+
2
(
5
−
1
)
(
3
−
1
)
16
{\displaystyle \cos \left({\frac {\pi }{60}}\right)=\cos(3^{\circ })={\frac {2{\big (}{\sqrt {3}}+1{\big )}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}{\big (}{\sqrt {5}}-1{\big )}{\big (}{\sqrt {3}}-1{\big )}}{16}}}
tan
(
π
60
)
=
tan
(
3
∘
)
=
(
(
2
−
3
)
(
3
+
5
)
−
2
)
(
2
−
10
−
2
5
)
4
{\displaystyle \tan \left({\frac {\pi }{60}}\right)=\tan(3^{\circ })={\frac {{\Big (}(2-{\sqrt {3}})(3+{\sqrt {5}})-2{\Big )}\left(2-{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}}
cot
(
π
60
)
=
cot
(
3
∘
)
=
(
(
2
+
3
)
(
3
+
5
)
−
2
)
(
2
+
10
−
2
5
)
4
{\displaystyle \cot \left({\frac {\pi }{60}}\right)=\cot(3^{\circ })={\frac {{\Big (}(2+{\sqrt {3}})(3+{\sqrt {5}})-2{\Big )}\left(2+{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}}
3.75°: מצולע בעל 48 צלעות
עריכה
sin
(
π
48
)
=
sin
(
3.75
∘
)
=
2
−
2
+
2
+
3
2
{\displaystyle \sin \left({\frac {\pi }{48}}\right)=\sin(3.75^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}
cos
(
π
48
)
=
cos
(
3.75
∘
)
=
2
+
2
+
2
+
3
2
{\displaystyle \cos \left({\frac {\pi }{48}}\right)=\cos(3.75^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}
4.5°: מצולע בעל 40 צלעות
עריכה
sin
(
π
40
)
=
sin
(
4.5
∘
)
=
(
4
−
2
2
)
(
5
+
5
)
−
(
5
−
1
)
2
+
2
8
{\displaystyle \sin \left({\frac {\pi }{40}}\right)=\sin(4.5^{\circ })={\frac {{\sqrt {(4-2{\sqrt {2}})(5+{\sqrt {5}})}}-{\big (}{\sqrt {5}}-1{\big )}{\sqrt {2+{\sqrt {2}}}}}{8}}}
cos
(
π
40
)
=
cos
(
4.5
∘
)
=
(
4
+
2
2
)
(
5
+
5
)
+
(
5
−
1
)
2
−
2
8
{\displaystyle \cos \left({\frac {\pi }{40}}\right)=\cos(4.5^{\circ })={\frac {{\sqrt {(4+2{\sqrt {2}})(5+{\sqrt {5}})}}+{\big (}{\sqrt {5}}-1{\big )}{\sqrt {2-{\sqrt {2}}}}}{8}}}
5.625°: מצולע בעל 32 צלעות
עריכה
sin
(
π
32
)
=
sin
(
5.625
∘
)
=
2
−
2
+
2
+
2
2
{\displaystyle \sin \left({\frac {\pi }{32}}\right)=\sin(5.625^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}}
cos
(
π
32
)
=
cos
(
5.625
∘
)
=
2
+
2
+
2
+
2
2
{\displaystyle \cos \left({\frac {\pi }{32}}\right)=\cos(5.625^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}}
6°: מצולע בעל 30 צלעות
עריכה
sin
(
π
30
)
=
sin
(
6
∘
)
=
30
−
6
5
−
5
−
1
8
{\displaystyle \sin \left({\frac {\pi }{30}}\right)=\sin(6^{\circ })={\frac {{\sqrt {30-6{\sqrt {5}}}}-{\sqrt {5}}-1}{8}}}
cos
(
π
30
)
=
cos
(
6
∘
)
=
10
−
2
5
+
3
+
15
8
{\displaystyle \cos \left({\frac {\pi }{30}}\right)=\cos(6^{\circ })={\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {3}}+{\sqrt {15}}}{8}}}
tan
(
π
30
)
=
tan
(
6
∘
)
=
10
−
2
5
+
3
−
15
2
{\displaystyle \tan \left({\frac {\pi }{30}}\right)=\tan(6^{\circ })={\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {3}}-{\sqrt {15}}}{2}}}
cot
(
π
30
)
=
cot
(
6
∘
)
=
3
3
+
15
+
50
+
22
5
2
{\displaystyle \cot \left({\frac {\pi }{30}}\right)=\cot(6^{\circ })={\frac {3{\sqrt {3}}+{\sqrt {15}}+{\sqrt {50+22{\sqrt {5}}}}}{2}}}
7.5°: מצולע בעל 24 צלעות
עריכה
sin
(
π
24
)
=
sin
(
7.5
∘
)
=
2
−
2
+
3
2
{\displaystyle \sin \left({\frac {\pi }{24}}\right)=\sin(7.5^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}{2}}}
cos
(
π
24
)
=
cos
(
7.5
∘
)
=
2
+
2
+
3
2
{\displaystyle \cos \left({\frac {\pi }{24}}\right)=\cos(7.5^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}{2}}}
tan
(
π
24
)
=
tan
(
7.5
∘
)
=
(
2
−
1
)
(
3
−
2
)
{\displaystyle \tan \left({\frac {\pi }{24}}\right)=\tan(7.5^{\circ })={\big (}{\sqrt {2}}-1{\big )}{\big (}{\sqrt {3}}-{\sqrt {2}}{\big )}}
cot
(
π
24
)
=
cot
(
7.5
∘
)
=
(
2
+
1
)
(
3
+
2
)
{\displaystyle \cot \left({\frac {\pi }{24}}\right)=\cot(7.5^{\circ })={\big (}{\sqrt {2}}+1{\big )}{\big (}{\sqrt {3}}+{\sqrt {2}}{\big )}}
9°: מצולע בעל 20 צלעות
עריכה
sin
(
π
20
)
=
sin
(
9
∘
)
=
10
+
2
−
2
5
−
5
8
{\displaystyle \sin \left({\frac {\pi }{20}}\right)=\sin(9^{\circ })={\frac {{\sqrt {10}}+{\sqrt {2}}-2{\sqrt {5-{\sqrt {5}}}}}{8}}}
cos
(
π
20
)
=
cos
(
9
∘
)
=
10
+
2
+
2
5
−
5
8
{\displaystyle \cos \left({\frac {\pi }{20}}\right)=\cos(9^{\circ })={\frac {{\sqrt {10}}+{\sqrt {2}}+2{\sqrt {5-{\sqrt {5}}}}}{8}}}
tan
(
π
20
)
=
tan
(
9
∘
)
=
5
+
1
−
5
+
2
5
{\displaystyle \tan \left({\frac {\pi }{20}}\right)=\tan(9^{\circ })={\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}}
cot
(
π
20
)
=
cot
(
9
∘
)
=
5
+
1
+
5
+
2
5
{\displaystyle \cot \left({\frac {\pi }{20}}\right)=\cot(9^{\circ })={\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}}
11.25°: מצולע בעל 16 צלעות
עריכה
sin
(
π
16
)
=
sin
(
11.25
∘
)
=
2
−
2
+
2
2
{\displaystyle \sin \left({\frac {\pi }{16}}\right)=\sin(11.25^{\circ })={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}{2}}}
cos
(
π
16
)
=
cos
(
11.25
∘
)
=
2
+
2
+
2
2
{\displaystyle \cos \left({\frac {\pi }{16}}\right)=\cos(11.25^{\circ })={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}}
tan
(
π
16
)
=
tan
(
11.25
∘
)
=
4
+
2
2
−
2
−
1
{\displaystyle \tan \left({\frac {\pi }{16}}\right)=\tan(11.25^{\circ })={\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2}}-1}
cot
(
π
16
)
=
cot
(
11.25
∘
)
=
4
+
2
2
+
2
+
1
{\displaystyle \cot \left({\frac {\pi }{16}}\right)=\cot(11.25^{\circ })={\sqrt {4+2{\sqrt {2}}}}+{\sqrt {2}}+1}
12°: מצולע בעל 15 צלעות
עריכה
sin
(
π
15
)
=
sin
(
12
∘
)
=
10
+
2
5
+
3
−
15
8
{\displaystyle \sin \left({\frac {\pi }{15}}\right)=\sin(12^{\circ })={\frac {{\sqrt {10+2{\sqrt {5}}}}+{\sqrt {3}}-{\sqrt {15}}}{8}}}
cos
(
π
15
)
=
cos
(
12
∘
)
=
30
+
6
5
+
5
−
1
8
{\displaystyle \cos \left({\frac {\pi }{15}}\right)=\cos(12^{\circ })={\frac {{\sqrt {30+6{\sqrt {5}}}}+{\sqrt {5}}-1}{8}}}
tan
(
π
15
)
=
tan
(
12
∘
)
=
3
3
−
15
−
50
−
22
5
2
{\displaystyle \tan \left({\frac {\pi }{15}}\right)=\tan(12^{\circ })={\frac {3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {50-22{\sqrt {5}}}}}{2}}}
cot
(
π
15
)
=
cot
(
12
∘
)
=
15
+
3
+
10
+
2
5
2
{\displaystyle \cot \left({\frac {\pi }{15}}\right)=\cot(12^{\circ })={\frac {{\sqrt {15}}+{\sqrt {3}}+{\sqrt {10+2{\sqrt {5}}}}}{2}}}
15°: מתורסר, מצולע בעל 12 צלעות
עריכה
sin
(
π
12
)
=
sin
(
15
∘
)
=
6
−
2
4
{\displaystyle \sin \left({\frac {\pi }{12}}\right)=\sin(15^{\circ })={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
cos
(
π
12
)
=
cos
(
15
∘
)
=
6
+
2
4
{\displaystyle \cos \left({\frac {\pi }{12}}\right)=\cos(15^{\circ })={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
tan
(
π
12
)
=
tan
(
15
∘
)
=
2
−
3
{\displaystyle \tan \left({\frac {\pi }{12}}\right)=\tan(15^{\circ })=2-{\sqrt {3}}}
cot
(
π
12
)
=
cot
(
15
∘
)
=
2
+
3
{\displaystyle \cot \left({\frac {\pi }{12}}\right)=\cot(15^{\circ })=2+{\sqrt {3}}}
18°: מעושר, מצולע בעל 10 צלעות
עריכה
sin
(
π
10
)
=
sin
(
18
∘
)
=
5
−
1
4
{\displaystyle \sin \left({\frac {\pi }{10}}\right)=\sin(18^{\circ })={\frac {{\sqrt {5}}-1}{4}}}
cos
(
π
10
)
=
cos
(
18
∘
)
=
10
+
2
5
4
{\displaystyle \cos \left({\frac {\pi }{10}}\right)=\cos(18^{\circ })={\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
tan
(
π
10
)
=
tan
(
18
∘
)
=
25
−
10
5
5
{\displaystyle \tan \left({\frac {\pi }{10}}\right)=\tan(18^{\circ })={\frac {\sqrt {25-10{\sqrt {5}}}}{5}}}
cot
(
π
10
)
=
cot
(
18
∘
)
=
5
+
2
5
{\displaystyle \cot \left({\frac {\pi }{10}}\right)=\cot(18^{\circ })={\sqrt {5+2{\sqrt {5}}}}}
סכום זויות: 21° = 9° + 12°
עריכה
sin
(
7
π
60
)
=
sin
(
21
∘
)
=
2
(
3
+
1
)
5
−
5
−
2
(
3
−
1
)
(
5
+
1
)
16
{\displaystyle \sin \left({\frac {7\pi }{60}}\right)=\sin(21^{\circ })={\frac {2{\big (}{\sqrt {3}}+1{\big )}{\sqrt {5-{\sqrt {5}}}}-{\sqrt {2}}{\big (}{\sqrt {3}}-1{\big )}{\big (}{\sqrt {5}}+1{\big )}}{16}}}
cos
(
7
π
60
)
=
cos
(
21
∘
)
=
2
(
3
−
1
)
5
−
5
+
2
(
3
+
1
)
(
5
+
1
)
16
{\displaystyle \cos \left({\frac {7\pi }{60}}\right)=\cos(21^{\circ })={\frac {2{\big (}{\sqrt {3}}-1{\big )}{\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}{\big (}{\sqrt {3}}+1{\big )}{\big (}{\sqrt {5}}+1{\big )}}{16}}}
tan
(
7
π
60
)
=
tan
(
21
∘
)
=
(
2
−
(
2
+
3
)
(
3
−
5
)
)
(
2
−
10
+
2
5
)
4
{\displaystyle \tan \left({\frac {7\pi }{60}}\right)=\tan(21^{\circ })={\frac {{\Big (}2-(2+{\sqrt {3}})(3-{\sqrt {5}}){\Big )}{\Big (}2-{\sqrt {10+2{\sqrt {5}}}}{\Big )}}{4}}}
cot
(
7
π
60
)
=
cot
(
21
∘
)
=
(
2
−
(
2
−
3
)
(
3
−
5
)
)
(
2
+
10
+
2
5
)
4
{\displaystyle \cot \left({\frac {7\pi }{60}}\right)=\cot(21^{\circ })={\frac {{\Big (}2-(2-{\sqrt {3}})(3-{\sqrt {5}}){\Big )}{\Big (}2+{\sqrt {10+2{\sqrt {5}}}}{\Big )}}{4}}}
22.5°: מתומן
עריכה
sin
(
π
8
)
=
sin
(
22.5
∘
)
=
2
−
2
2
{\displaystyle \sin \left({\frac {\pi }{8}}\right)=\sin(22.5^{\circ })={\frac {\sqrt {2-{\sqrt {2}}}}{2}}}
cos
(
π
8
)
=
cos
(
22.5
∘
)
=
2
+
2
2
{\displaystyle \cos \left({\frac {\pi }{8}}\right)=\cos(22.5^{\circ })={\frac {\sqrt {2+{\sqrt {2}}}}{2}}}
tan
(
π
8
)
=
tan
(
22.5
∘
)
=
2
−
1
{\displaystyle \tan \left({\frac {\pi }{8}}\right)=\tan(22.5^{\circ })={\sqrt {2}}-1}
cot
(
π
8
)
=
cot
(
22.5
∘
)
=
2
+
1
{\displaystyle \cot \left({\frac {\pi }{8}}\right)=\cot(22.5^{\circ })={\sqrt {2}}+1}
סכום זויות: 24° = 12° + 12°
עריכה
sin
(
2
π
15
)
=
sin
(
24
∘
)
=
15
+
3
−
10
−
2
5
8
{\displaystyle \sin \left({\frac {2\pi }{15}}\right)=\sin(24^{\circ })={\frac {{\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}}{8}}}
cos
(
2
π
15
)
=
cos
(
24
∘
)
=
30
−
6
5
+
5
+
1
8
{\displaystyle \cos \left({\frac {2\pi }{15}}\right)=\cos(24^{\circ })={\frac {{\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1}{8}}}
tan
(
2
π
15
)
=
tan
(
24
∘
)
=
50
+
22
5
−
3
3
−
15
2
{\displaystyle \tan \left({\frac {2\pi }{15}}\right)=\tan(24^{\circ })={\frac {{\sqrt {50+22{\sqrt {5}}}}-3{\sqrt {3}}-{\sqrt {15}}}{2}}}
cot
(
2
π
15
)
=
cot
(
24
∘
)
=
15
−
3
+
10
−
2
5
2
{\displaystyle \cot \left({\frac {2\pi }{15}}\right)=\cot(24^{\circ })={\frac {{\sqrt {15}}-{\sqrt {3}}+{\sqrt {10-2{\sqrt {5}}}}}{2}}}
סכום זויות: 27° = 12° + 15°
עריכה
sin
(
3
π
20
)
=
sin
(
27
∘
)
=
2
5
+
5
−
2
(
5
−
1
)
8
{\displaystyle \sin \left({\frac {3\pi }{20}}\right)=\sin(27^{\circ })={\frac {2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {5}}-1)}{8}}}
cos
(
3
π
20
)
=
cos
(
27
∘
)
=
2
5
+
5
+
2
(
5
−
1
)
8
{\displaystyle \cos \left({\frac {3\pi }{20}}\right)=\cos(27^{\circ })={\frac {2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {5}}-1)}{8}}}
tan
(
3
π
20
)
=
tan
(
27
∘
)
=
5
−
1
−
5
−
2
5
{\displaystyle \tan \left({\frac {3\pi }{20}}\right)=\tan(27^{\circ })={\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}}
cot
(
3
π
20
)
=
cot
(
27
∘
)
=
5
−
1
+
5
−
2
5
{\displaystyle \cot \left({\frac {3\pi }{20}}\right)=\cot(27^{\circ })={\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}}
30°: משושה
עריכה
sin
(
π
6
)
=
sin
(
30
∘
)
=
1
2
{\displaystyle \sin \left({\frac {\pi }{6}}\right)=\sin(30^{\circ })={\frac {1}{2}}}
cos
(
π
6
)
=
cos
(
30
∘
)
=
3
2
{\displaystyle \cos \left({\frac {\pi }{6}}\right)=\cos(30^{\circ })={\frac {\sqrt {3}}{2}}}
tan
(
π
6
)
=
tan
(
30
∘
)
=
3
3
=
3
3
{\displaystyle \tan \left({\frac {\pi }{6}}\right)=\tan(30^{\circ })={\frac {\sqrt {3}}{3}}={\frac {\sqrt {3}}{3}}}
cot
(
π
6
)
=
cot
(
30
∘
)
=
3
{\displaystyle \cot \left({\frac {\pi }{6}}\right)=\cot(30^{\circ })={\sqrt {3}}}
סכום זויות: 33° = 15° + 18°
עריכה
sin
(
11
π
60
)
=
sin
(
33
∘
)
=
2
(
3
−
1
)
5
+
5
+
2
(
3
+
1
)
(
5
−
1
)
16
{\displaystyle \sin \left({\frac {11\pi }{60}}\right)=\sin(33^{\circ })={\frac {2{\big (}{\sqrt {3}}-1{\big )}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}{\big (}{\sqrt {3}}+1{\big )}{\big (}{\sqrt {5}}-1{\big )}}{16}}}
cos
(
11
π
60
)
=
cos
(
33
∘
)
=
2
(
3
+
1
)
5
+
5
−
2
(
3
−
1
)
(
5
−
1
)
16
{\displaystyle \cos \left({\frac {11\pi }{60}}\right)=\cos(33^{\circ })={\frac {2{\big (}{\sqrt {3}}+1{\big )}{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}{\big (}{\sqrt {3}}-1{\big )}{\big (}{\sqrt {5}}-1{\big )}}{16}}}
tan
(
11
π
60
)
=
tan
(
33
∘
)
=
(
2
−
(
2
−
3
)
(
3
+
5
)
)
(
2
+
10
−
2
5
)
4
{\displaystyle \tan \left({\frac {11\pi }{60}}\right)=\tan(33^{\circ })={\frac {{\Big (}2-(2-{\sqrt {3}})(3+{\sqrt {5}}){\Big )}\left(2+{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}}
cot
(
11
π
60
)
=
cot
(
33
∘
)
=
(
2
−
(
2
+
3
)
(
3
+
5
)
)
(
2
−
10
−
2
5
)
4
{\displaystyle \cot \left({\frac {11\pi }{60}}\right)=\cot(33^{\circ })={\frac {{\Big (}2-(2+{\sqrt {3}})(3+{\sqrt {5}}){\Big )}\left(2-{\sqrt {10-2{\sqrt {5}}}}\right)}{4}}}
36°: מחומש
עריכה
sin
(
π
5
)
=
sin
(
36
∘
)
=
10
−
2
5
4
{\displaystyle \sin \left({\frac {\pi }{5}}\right)=\sin(36^{\circ })={\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
cos
(
π
5
)
=
cos
(
36
∘
)
=
1
+
5
4
{\displaystyle \cos \left({\frac {\pi }{5}}\right)=\cos(36^{\circ })={\frac {1+{\sqrt {5}}}{4}}}
tan
(
π
5
)
=
tan
(
36
∘
)
=
5
−
2
5
{\displaystyle \tan \left({\frac {\pi }{5}}\right)=\tan(36^{\circ })={\sqrt {5-2{\sqrt {5}}}}}
cot
(
π
5
)
=
cot
(
36
∘
)
=
25
+
10
5
5
{\displaystyle \cot \left({\frac {\pi }{5}}\right)=\cot(36^{\circ })={\frac {\sqrt {25+10{\sqrt {5}}}}{5}}}
סכום זויות: 39° = 18° + 21°
עריכה
sin
(
13
π
60
)
=
sin
(
39
∘
)
=
2
(
3
+
1
)
(
5
+
1
)
−
2
(
3
−
1
)
5
−
5
16
{\displaystyle \sin \left({\frac {13\pi }{60}}\right)=\sin(39^{\circ })={\frac {{\sqrt {2}}{\big (}{\sqrt {3}}+1{\big )}{\big (}{\sqrt {5}}+1{\big )}-2{\big (}{\sqrt {3}}-1{\big )}{\sqrt {5-{\sqrt {5}}}}}{16}}}
cos
(
13
π
60
)
=
cos
(
39
∘
)
=
2
(
3
+
1
)
5
−
5
+
2
(
3
−
1
)
(
5
+
1
)
16
{\displaystyle \cos \left({\frac {13\pi }{60}}\right)=\cos(39^{\circ })={\frac {2{\big (}{\sqrt {3}}+1{\big )}{\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}{\big (}{\sqrt {3}}-1{\big )}{\big (}{\sqrt {5}}+1{\big )}}{16}}}
tan
(
13
π
60
)
=
tan
(
39
∘
)
=
(
(
2
−
3
)
(
3
−
5
)
−
2
)
(
2
−
10
+
2
5
)
4
{\displaystyle \tan \left({\frac {13\pi }{60}}\right)=\tan(39^{\circ })={\frac {{\Big (}(2-{\sqrt {3}})(3-{\sqrt {5}})-2{\Big )}\left(2-{\sqrt {10+2{\sqrt {5}}}}\right)}{4}}}
cot
(
13
π
60
)
=
cot
(
39
∘
)
=
(
(
2
+
3
)
(
3
−
5
)
−
2
)
(
2
+
10
+
2
5
)
4
{\displaystyle \cot \left({\frac {13\pi }{60}}\right)=\cot(39^{\circ })={\frac {{\Big (}(2+{\sqrt {3}})(3-{\sqrt {5}})-2{\Big )}\left(2+{\sqrt {10+2{\sqrt {5}}}}\right)}{4}}}
סכום זויות: 42° = 21° + 21°
עריכה
sin
(
7
π
30
)
=
sin
(
42
∘
)
=
30
+
6
5
−
5
+
1
8
{\displaystyle \sin \left({\frac {7\pi }{30}}\right)=\sin(42^{\circ })={\frac {{\sqrt {30+6{\sqrt {5}}}}-{\sqrt {5}}+1}{8}}}
cos
(
7
π
30
)
=
cos
(
42
∘
)
=
15
−
3
+
10
+
2
5
8
{\displaystyle \cos \left({\frac {7\pi }{30}}\right)=\cos(42^{\circ })={\frac {{\sqrt {15}}-{\sqrt {3}}+{\sqrt {10+2{\sqrt {5}}}}}{8}}}
tan
(
7
π
30
)
=
tan
(
42
∘
)
=
15
+
3
−
10
+
2
5
2
{\displaystyle \tan \left({\frac {7\pi }{30}}\right)=\tan(42^{\circ })={\frac {{\sqrt {15}}+{\sqrt {3}}-{\sqrt {10+2{\sqrt {5}}}}}{2}}}
cot
(
7
π
30
)
=
cot
(
42
∘
)
=
50
−
22
5
+
3
3
−
15
2
{\displaystyle \cot \left({\frac {7\pi }{30}}\right)=\cot(42^{\circ })={\frac {{\sqrt {50-22{\sqrt {5}}}}+3{\sqrt {3}}-{\sqrt {15}}}{2}}}
45°: ריבוע
עריכה
sin
(
π
4
)
=
sin
(
45
∘
)
=
2
2
{\displaystyle \sin \left({\frac {\pi }{4}}\right)=\sin(45^{\circ })={\frac {\sqrt {2}}{2}}}
cos
(
π
4
)
=
cos
(
45
∘
)
=
2
2
{\displaystyle \cos \left({\frac {\pi }{4}}\right)=\cos(45^{\circ })={\frac {\sqrt {2}}{2}}}
tan
(
π
4
)
=
tan
(
45
∘
)
=
1
{\displaystyle \tan \left({\frac {\pi }{4}}\right)=\tan(45^{\circ })=1}
cot
(
π
4
)
=
cot
(
45
∘
)
=
1
{\displaystyle \cot \left({\frac {\pi }{4}}\right)=\cot(45^{\circ })=1}
סכום זויות: 54° = 27° + 27°
עריכה
sin
(
3
π
10
)
=
sin
(
54
∘
)
=
1
+
5
4
{\displaystyle \sin \left({\frac {3\pi }{10}}\right)=\sin(54^{\circ })={\frac {1+{\sqrt {5}}}{4}}}
cos
(
3
π
10
)
=
cos
(
54
∘
)
=
10
−
2
5
4
{\displaystyle \cos \left({\frac {3\pi }{10}}\right)=\cos(54^{\circ })={\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
tan
(
3
π
10
)
=
tan
(
54
∘
)
=
25
+
10
5
5
{\displaystyle \tan \left({\frac {3\pi }{10}}\right)=\tan(54^{\circ })={\frac {\sqrt {25+10{\sqrt {5}}}}{5}}}
cot
(
3
π
10
)
=
cot
(
54
∘
)
=
5
−
2
5
{\displaystyle \cot \left({\frac {3\pi }{10}}\right)=\cot(54^{\circ })={\sqrt {5-2{\sqrt {5}}}}}
60°: משולש שוה-צלעות
עריכה
sin
(
π
3
)
=
sin
(
60
∘
)
=
3
2
{\displaystyle \sin \left({\frac {\pi }{3}}\right)=\sin(60^{\circ })={\frac {\sqrt {3}}{2}}}
cos
(
π
3
)
=
cos
(
60
∘
)
=
1
2
{\displaystyle \cos \left({\frac {\pi }{3}}\right)=\cos(60^{\circ })={\frac {1}{2}}}
tan
(
π
3
)
=
tan
(
60
∘
)
=
3
{\displaystyle \tan \left({\frac {\pi }{3}}\right)=\tan(60^{\circ })={\sqrt {3}}}
cot
(
π
3
)
=
cot
(
60
∘
)
=
3
3
{\displaystyle \cot \left({\frac {\pi }{3}}\right)=\cot(60^{\circ })={\frac {\sqrt {3}}{3}}}
סכום זויות: 67.5° = 7.5° + 60°
עריכה
sin
(
3
π
8
)
=
sin
(
67.5
∘
)
=
2
+
2
2
{\displaystyle \sin \left({\frac {3\pi }{8}}\right)=\sin(67.5^{\circ })={\frac {\sqrt {2+{\sqrt {2}}}}{2}}}
cos
(
3
π
8
)
=
cos
(
67.5
∘
)
=
2
−
2
2
{\displaystyle \cos \left({\frac {3\pi }{8}}\right)=\cos(67.5^{\circ })={\frac {\sqrt {2-{\sqrt {2}}}}{2}}}
tan
(
3
π
8
)
=
tan
(
67.5
∘
)
=
2
+
1
{\displaystyle \tan \left({\frac {3\pi }{8}}\right)=\tan(67.5^{\circ })={\sqrt {2}}+1}
cot
(
3
π
8
)
=
cot
(
67.5
∘
)
=
2
−
1
{\displaystyle \cot \left({\frac {3\pi }{8}}\right)=\cot(67.5^{\circ })={\sqrt {2}}-1}
סכום זויות: 72° = 36° + 36°
עריכה
sin
(
2
π
5
)
=
sin
(
72
∘
)
=
10
+
2
5
4
{\displaystyle \sin \left({\frac {2\pi }{5}}\right)=\sin(72^{\circ })={\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
cos
(
2
π
5
)
=
cos
(
72
∘
)
=
5
−
1
4
{\displaystyle \cos \left({\frac {2\pi }{5}}\right)=\cos(72^{\circ })={\frac {{\sqrt {5}}-1}{4}}}
tan
(
2
π
5
)
=
tan
(
72
∘
)
=
5
+
2
5
{\displaystyle \tan \left({\frac {2\pi }{5}}\right)=\tan(72^{\circ })={\sqrt {5+2{\sqrt {5}}}}}
cot
(
2
π
5
)
=
cot
(
72
∘
)
=
25
−
10
5
5
{\displaystyle \cot \left({\frac {2\pi }{5}}\right)=\cot(72^{\circ })={\frac {\sqrt {25-10{\sqrt {5}}}}{5}}}
סכום זויות: 75° = 30° + 45°
עריכה
sin
(
5
π
12
)
=
sin
(
75
∘
)
=
6
+
2
4
{\displaystyle \sin \left({\frac {5\pi }{12}}\right)=\sin(75^{\circ })={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}}
cos
(
5
π
12
)
=
cos
(
75
∘
)
=
6
−
2
4
{\displaystyle \cos \left({\frac {5\pi }{12}}\right)=\cos(75^{\circ })={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}}
tan
(
5
π
12
)
=
tan
(
75
∘
)
=
2
+
3
{\displaystyle \tan \left({\frac {5\pi }{12}}\right)=\tan(75^{\circ })=2+{\sqrt {3}}}
cot
(
5
π
12
)
=
cot
(
75
∘
)
=
2
−
3
{\displaystyle \cot \left({\frac {5\pi }{12}}\right)=\cot(75^{\circ })=2-{\sqrt {3}}}
90°: זוית ישרה
עריכה
sin
(
π
2
)
=
sin
(
90
∘
)
=
1
{\displaystyle \sin \left({\frac {\pi }{2}}\right)=\sin(90^{\circ })=1}
cos
(
π
2
)
=
cos
(
90
∘
)
=
0
{\displaystyle \cos \left({\frac {\pi }{2}}\right)=\cos(90^{\circ })=0}
tan
(
π
2
)
=
tan
(
90
∘
)
=
undefined
{\displaystyle \tan \left({\frac {\pi }{2}}\right)=\tan(90^{\circ })={\text{ undefined}}}
cot
(
π
2
)
=
cot
(
90
∘
)
=
0
{\displaystyle \cot \left({\frac {\pi }{2}}\right)=\cot(90^{\circ })=0}
