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yOuR tExT
#0°: fundamental
#90°: fundamental
Further angles Exact trigonometric table for multiples of 3 degrees. Values outside the [0°, 45°] angle range are trivially derived from these values, using circle axis
.)
In the entries below, when a certain number of degrees is related to a regular polygon, the relation is that the number of degrees in each angle of the polygon is (n – 2) times the indicated number of degrees (where n is the number of sides). This is because the sum of the angles of any n -gon is 180° × (n – 2) and so the measure of each angle of any regular n -gon is 180° × (n – 2) ÷ n . Thus for example the entry "45°: square" means that, with n = 4, 180° ÷ n = 45°, and the number of degrees in each angle of a square is (n – 2) × 45° = 90°.
sin
0
=
cos
π
2
=
cos
90
∘
=
0
{\displaystyle \sin 0=\cos {\frac {\pi }{2}}=\cos 90^{\circ }=0\,}
cos
0
=
sin
π
2
=
sin
90
∘
=
1
{\displaystyle \cos 0=\sin {\frac {\pi }{2}}=\sin 90^{\circ }=1\,}
tan
0
=
cot
π
2
=
cot
90
∘
=
0
{\displaystyle \tan 0=\cot {\frac {\pi }{2}}=\cot 90^{\circ }=0\,}
cot
0
=
tan
π
2
=
tan
90
∘
is undefined
{\displaystyle \cot 0=\tan {\frac {\pi }{2}}=\tan 90^{\circ }{\text{ is undefined}}\,}
1.5°: regular hecatonicosagon (120-sided polygon)
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 \left(1.5^{\circ }\right)={\frac {\left({\sqrt {2+{\sqrt {2}}}}\right)\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)-\left({\sqrt {2-{\sqrt {2}}}}\right)\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 \left(1.5^{\circ }\right)={\frac {\left({\sqrt {2+{\sqrt {2}}}}\right)\left({\sqrt {30-6{\sqrt {5}}}}+{\sqrt {5}}+1\right)+\left({\sqrt {2-{\sqrt {2}}}}\right)\left({\sqrt {15}}+{\sqrt {3}}-{\sqrt {10-2{\sqrt {5}}}}\right)}{16}}}
1.875°: regular enneacontahexagon (96-sided polygon)
sin
(
π
96
)
=
sin
(
1.875
∘
)
=
1
2
2
−
2
+
2
+
2
+
3
{\displaystyle \sin \left({\frac {\pi }{96}}\right)=\sin \left(1.875^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}}
cos
(
π
96
)
=
cos
(
1.875
∘
)
=
1
2
2
+
2
+
2
+
2
+
3
{\displaystyle \cos \left({\frac {\pi }{96}}\right)=\cos \left(1.875^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}}}
2.25°: regular octacontagon (80-sided polygon)
sin
(
π
80
)
=
sin
(
2.25
∘
)
=
1
2
2
−
2
+
2
+
5
+
5
2
{\displaystyle \sin \left({\frac {\pi }{80}}\right)=\sin \left(2.25^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}}}
cos
(
π
80
)
=
cos
(
2.25
∘
)
=
1
2
2
+
2
+
2
+
5
+
5
2
{\displaystyle \cos \left({\frac {\pi }{80}}\right)=\cos \left(2.25^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}}}
2.8125°: regular hexacontatetragon (64-sided polygon)
sin
(
π
64
)
=
sin
(
2.8125
∘
)
=
1
2
2
−
2
+
2
+
2
+
2
{\displaystyle \sin \left({\frac {\pi }{64}}\right)=\sin \left(2.8125^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}}
cos
(
π
64
)
=
cos
(
2.8125
∘
)
=
1
2
2
+
2
+
2
+
2
+
2
{\displaystyle \cos \left({\frac {\pi }{64}}\right)=\cos \left(2.8125^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}}}
3°: regular hexacontagon (60-sided polygon)
sin
(
π
60
)
=
sin
(
3
∘
)
=
2
(
1
−
3
)
5
+
5
+
(
10
−
2
)
(
3
+
1
)
16
{\displaystyle \sin \left({\frac {\pi }{60}}\right)=\sin \left(3^{\circ }\right)={\frac {2\left(1-{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}+1\right)}{16}}\,}
cos
(
π
60
)
=
cos
(
3
∘
)
=
2
(
1
+
3
)
5
+
5
+
(
10
−
2
)
(
3
−
1
)
16
{\displaystyle \cos \left({\frac {\pi }{60}}\right)=\cos \left(3^{\circ }\right)={\frac {2\left(1+{\sqrt {3}}\right){\sqrt {5+{\sqrt {5}}}}+\left({\sqrt {10}}-{\sqrt {2}}\right)\left({\sqrt {3}}-1\right)}{16}}\,}
tan
(
π
60
)
=
tan
(
3
∘
)
=
[
(
2
−
3
)
(
3
+
5
)
−
2
]
[
2
−
10
−
2
5
]
4
{\displaystyle \tan \left({\frac {\pi }{60}}\right)=\tan \left(3^{\circ }\right)={\frac {\left[\left(2-{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)-2\right]\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 \left(3^{\circ }\right)={\frac {\left[\left(2+{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)-2\right]\left[2+{\sqrt {10-2{\sqrt {5}}}}\right]}{4}}\,}
3.75°: regular tetracontaoctagon (48-sided polygon)
sin
(
π
48
)
=
sin
(
3.75
∘
)
=
1
2
2
−
2
+
2
+
3
{\displaystyle \sin \left({\frac {\pi }{48}}\right)=\sin \left(3.75^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}
cos
(
π
48
)
=
cos
(
3.75
∘
)
=
1
2
2
+
2
+
2
+
3
{\displaystyle \cos \left({\frac {\pi }{48}}\right)=\cos \left(3.75^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}}
4.5°: regular tetracontagon (40-sided polygon)
sin
(
π
40
)
=
sin
(
4.5
∘
)
=
1
2
2
−
2
+
5
+
5
2
{\displaystyle \sin \left({\frac {\pi }{40}}\right)=\sin \left(4.5^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}
cos
(
π
40
)
=
cos
(
4.5
∘
)
=
1
2
2
+
2
+
5
+
5
2
{\displaystyle \cos \left({\frac {\pi }{40}}\right)=\cos \left(4.5^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}}}
5.625°: regular triacontadigon (32-sided polygon)
sin
(
π
32
)
=
sin
(
5.625
∘
)
=
1
2
2
−
2
+
2
+
2
{\displaystyle \sin \left({\frac {\pi }{32}}\right)=\sin \left(5.625^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}
cos
(
π
32
)
=
cos
(
5.625
∘
)
=
1
2
2
+
2
+
2
+
2
{\displaystyle \cos \left({\frac {\pi }{32}}\right)=\cos \left(5.625^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}
6°: regular triacontagon (30-sided polygon)
sin
π
30
=
sin
6
∘
=
30
−
180
−
5
−
1
8
{\displaystyle \sin {\frac {\pi }{30}}=\sin 6^{\circ }={\frac {{\sqrt {30-{\sqrt {180}}}}-{\sqrt {5}}-1}{8}}\,}
cos
π
30
=
cos
6
∘
=
10
−
20
+
3
+
15
8
{\displaystyle \cos {\frac {\pi }{30}}=\cos 6^{\circ }={\frac {{\sqrt {10-{\sqrt {20}}}}+{\sqrt {3}}+{\sqrt {15}}}{8}}\,}
tan
π
30
=
tan
6
∘
=
10
−
20
+
3
−
15
2
{\displaystyle \tan {\frac {\pi }{30}}=\tan 6^{\circ }={\frac {{\sqrt {10-{\sqrt {20}}}}+{\sqrt {3}}-{\sqrt {15}}}{2}}\,}
cot
π
30
=
cot
6
∘
=
27
+
15
+
50
+
2420
2
{\displaystyle \cot {\frac {\pi }{30}}=\cot 6^{\circ }={\frac {{\sqrt {27}}+{\sqrt {15}}+{\sqrt {50+{\sqrt {2420}}}}}{2}}\,}
7.5°: regular icositetragon (24-sided polygon)
sin
(
π
24
)
=
sin
(
7.5
∘
)
=
1
2
2
−
2
+
3
=
1
4
8
−
2
6
−
2
2
{\displaystyle \sin \left({\frac {\pi }{24}}\right)=\sin \left(7.5^{\circ }\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}={\frac {1}{4}}{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}}
cos
(
π
24
)
=
cos
(
7.5
∘
)
=
1
2
2
+
2
+
3
=
1
4
8
+
2
6
+
2
2
{\displaystyle \cos \left({\frac {\pi }{24}}\right)=\cos \left(7.5^{\circ }\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}={\frac {1}{4}}{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}}
tan
(
π
24
)
=
tan
(
7.5
∘
)
=
6
−
3
+
2
−
2
=
(
2
−
1
)
(
3
−
2
)
{\displaystyle \tan \left({\frac {\pi }{24}}\right)=\tan \left(7.5^{\circ }\right)={\sqrt {6}}-{\sqrt {3}}+{\sqrt {2}}-2\ =\left({\sqrt {2}}-1\right)\left({\sqrt {3}}-{\sqrt {2}}\right)}
cot
(
π
24
)
=
cot
(
7.5
∘
)
=
6
+
3
+
2
+
2
=
(
2
+
1
)
(
3
+
2
)
{\displaystyle \cot \left({\frac {\pi }{24}}\right)=\cot \left(7.5^{\circ }\right)={\sqrt {6}}+{\sqrt {3}}+{\sqrt {2}}+2\ =\left({\sqrt {2}}+1\right)\left({\sqrt {3}}+{\sqrt {2}}\right)}
9°: regular icosagon (20-sided polygon)
sin
π
20
=
sin
9
∘
=
1
2
2
−
5
+
5
2
{\displaystyle \sin {\frac {\pi }{20}}=\sin 9^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}
cos
π
20
=
cos
9
∘
=
1
2
2
+
5
+
5
2
{\displaystyle \cos {\frac {\pi }{20}}=\cos 9^{\circ }={\frac {1}{2}}{\sqrt {2+{\sqrt {\frac {5+{\sqrt {5}}}{2}}}}}}
tan
π
20
=
tan
9
∘
=
5
+
1
−
5
+
2
5
{\displaystyle \tan {\frac {\pi }{20}}=\tan 9^{\circ }={\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}\,}
cot
π
20
=
cot
9
∘
=
5
+
1
+
5
+
2
5
{\displaystyle \cot {\frac {\pi }{20}}=\cot 9^{\circ }={\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}\,}
11.25°: regular hexadecagon (16-sided polygon)
sin
π
16
=
sin
11.25
∘
=
1
2
2
−
2
+
2
{\displaystyle \sin {\frac {\pi }{16}}=\sin 11.25^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}
cos
π
16
=
cos
11.25
∘
=
1
2
2
+
2
+
2
{\displaystyle \cos {\frac {\pi }{16}}=\cos 11.25^{\circ }={\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}
tan
π
16
=
tan
11.25
∘
=
4
+
2
2
−
2
−
1
{\displaystyle \tan {\frac {\pi }{16}}=\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 {\frac {\pi }{16}}=\cot 11.25^{\circ }={\sqrt {4+2{\sqrt {2}}}}+{\sqrt {2}}+1}
12°: regular pentadecagon (15-sided polygon)
sin
π
15
=
sin
12
∘
=
1
8
[
2
(
5
+
5
)
+
3
−
15
]
{\displaystyle \sin {\frac {\pi }{15}}=\sin 12^{\circ }={\tfrac {1}{8}}\left[{\sqrt {2\left(5+{\sqrt {5}}\right)}}+{\sqrt {3}}-{\sqrt {15}}\right]\,}
cos
π
15
=
cos
12
∘
=
1
8
[
6
(
5
+
5
)
+
5
−
1
]
{\displaystyle \cos {\frac {\pi }{15}}=\cos 12^{\circ }={\tfrac {1}{8}}\left[{\sqrt {6\left(5+{\sqrt {5}}\right)}}+{\sqrt {5}}-1\right]\,}
tan
π
15
=
tan
12
∘
=
1
2
[
3
3
−
15
−
2
(
25
−
11
5
)
]
{\displaystyle \tan {\frac {\pi }{15}}=\tan 12^{\circ }={\tfrac {1}{2}}\left[3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {2\left(25-11{\sqrt {5}}\right)}}\,\right]\,}
cot
π
15
=
cot
12
∘
=
1
2
[
15
+
3
+
2
(
5
+
5
)
]
{\displaystyle \cot {\frac {\pi }{15}}=\cot 12^{\circ }={\tfrac {1}{2}}\left[{\sqrt {15}}+{\sqrt {3}}+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}
15°: regular dodecagon (12-sided polygon)
sin
π
12
=
sin
15
∘
=
1
4
(
6
−
2
)
=
1
2
2
−
3
{\displaystyle \sin {\frac {\pi }{12}}=\sin 15^{\circ }={\frac {1}{4}}\left({\sqrt {6}}-{\sqrt {2}}\right)={\frac {1}{2}}{\sqrt {2-{\sqrt {3}}}}}
cos
π
12
=
cos
15
∘
=
1
4
(
6
+
2
)
=
1
2
2
+
3
{\displaystyle \cos {\frac {\pi }{12}}=\cos 15^{\circ }={\frac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)={\frac {1}{2}}{\sqrt {2+{\sqrt {3}}}}}
tan
π
12
=
tan
15
∘
=
2
−
3
{\displaystyle \tan {\frac {\pi }{12}}=\tan 15^{\circ }=2-{\sqrt {3}}\,}
cot
π
12
=
cot
15
∘
=
2
+
3
{\displaystyle \cot {\frac {\pi }{12}}=\cot 15^{\circ }=2+{\sqrt {3}}\,}
75°: sum 30° + 45°
sin
5
π
12
=
sin
75
∘
=
1
4
(
6
+
2
)
{\displaystyle \sin {\frac {5\pi }{12}}=\sin 75^{\circ }={\tfrac {1}{4}}\left({\sqrt {6}}+{\sqrt {2}}\right)\,}
cos
5
π
12
=
cos
75
∘
=
1
4
(
6
−
2
)
{\displaystyle \cos {\frac {5\pi }{12}}=\cos 75^{\circ }={\tfrac {1}{4}}\left({\sqrt {6}}-{\sqrt {2}}\right)\,}
tan
5
π
12
=
tan
75
∘
=
2
+
3
{\displaystyle \tan {\frac {5\pi }{12}}=\tan 75^{\circ }=2+{\sqrt {3}}\,}
cot
5
π
12
=
cot
75
∘
=
2
−
3
{\displaystyle \cot {\frac {5\pi }{12}}=\cot 75^{\circ }=2-{\sqrt {3}}\,}
18°: regular decagon (10-sided polygon)[1]
sin
π
10
=
sin
18
∘
=
1
4
(
5
−
1
)
=
1
1
+
5
{\displaystyle \sin {\frac {\pi }{10}}=\sin 18^{\circ }={\tfrac {1}{4}}\left({\sqrt {5}}-1\right)={\frac {1}{1+{\sqrt {5}}}}\,}
cos
π
10
=
cos
18
∘
=
1
4
2
(
5
+
5
)
{\displaystyle \cos {\frac {\pi }{10}}=\cos 18^{\circ }={\tfrac {1}{4}}{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,}
tan
π
10
=
tan
18
∘
=
1
5
5
(
5
−
2
5
)
{\displaystyle \tan {\frac {\pi }{10}}=\tan 18^{\circ }={\tfrac {1}{5}}{\sqrt {5\left(5-2{\sqrt {5}}\right)}}\,}
cot
π
10
=
cot
18
∘
=
5
+
2
5
{\displaystyle \cot {\frac {\pi }{10}}=\cot 18^{\circ }={\sqrt {5+2{\sqrt {5}}}}\,}
72°: sum 36° + 36°
sin
2
π
5
=
sin
72
∘
=
1
4
2
(
5
+
5
)
{\displaystyle \sin {\frac {2\pi }{5}}=\sin 72^{\circ }={\tfrac {1}{4}}{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,}
cos
2
π
5
=
cos
72
∘
=
1
4
(
5
−
1
)
{\displaystyle \cos {\frac {2\pi }{5}}=\cos 72^{\circ }={\tfrac {1}{4}}\left({\sqrt {5}}-1\right)\,}
tan
2
π
5
=
tan
72
∘
=
5
+
2
5
{\displaystyle \tan {\frac {2\pi }{5}}=\tan 72^{\circ }={\sqrt {5+2{\sqrt {5}}}}\,}
cot
2
π
5
=
cot
72
∘
=
1
5
5
(
5
−
2
5
)
{\displaystyle \cot {\frac {2\pi }{5}}=\cot 72^{\circ }={\tfrac {1}{5}}{\sqrt {5\left(5-2{\sqrt {5}}\right)}}\,}
21°: sum 9° + 12°
sin
7
π
60
=
sin
21
∘
=
1
16
(
2
(
3
+
1
)
5
−
5
−
(
6
−
2
)
(
1
+
5
)
)
{\displaystyle \sin {\frac {7\pi }{60}}=\sin 21^{\circ }={\frac {1}{16}}\left(2\left({\sqrt {3}}+1\right){\sqrt {5-{\sqrt {5}}}}-\left({\sqrt {6}}-{\sqrt {2}}\right)\left(1+{\sqrt {5}}\right)\right)\,}
cos
7
π
60
=
cos
21
∘
=
1
16
(
2
(
3
−
1
)
5
−
5
+
(
6
+
2
)
(
1
+
5
)
)
{\displaystyle \cos {\frac {7\pi }{60}}=\cos 21^{\circ }={\frac {1}{16}}\left(2\left({\sqrt {3}}-1\right){\sqrt {5-{\sqrt {5}}}}+\left({\sqrt {6}}+{\sqrt {2}}\right)\left(1+{\sqrt {5}}\right)\right)\,}
tan
7
π
60
=
tan
21
∘
=
1
4
(
2
−
(
2
+
3
)
(
3
−
5
)
)
(
2
−
2
(
5
+
5
)
)
{\displaystyle \tan {\frac {7\pi }{60}}=\tan 21^{\circ }={\frac {1}{4}}\left(2-\left(2+{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)\right)\left(2-{\sqrt {2\left(5+{\sqrt {5}}\right)}}\right)\,}
cot
7
π
60
=
cot
21
∘
=
1
4
(
2
−
(
2
−
3
)
(
3
−
5
)
)
(
2
+
2
(
5
+
5
)
)
{\displaystyle \cot {\frac {7\pi }{60}}=\cot 21^{\circ }={\frac {1}{4}}\left(2-\left(2-{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)\right)\left(2+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\right)\,}
22.5°: regular octagon
sin
π
8
=
sin
22.5
∘
=
1
2
2
−
2
,
{\displaystyle \sin {\frac {\pi }{8}}=\sin 22.5^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {2}}}},}
cos
π
8
=
cos
22.5
∘
=
1
2
2
+
2
{\displaystyle \cos {\frac {\pi }{8}}=\cos 22.5^{\circ }={\frac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\,}
tan
π
8
=
tan
22.5
∘
=
2
−
1
{\displaystyle \tan {\frac {\pi }{8}}=\tan 22.5^{\circ }={\sqrt {2}}-1\,}
cot
π
8
=
cot
22.5
∘
=
2
+
1
=
δ
S
{\displaystyle \cot {\frac {\pi }{8}}=\cot 22.5^{\circ }={\sqrt {2}}+1=\delta _{S}\,}
silver ratio 67.5°: sum 7.5° + 60°
sin
3
π
8
=
sin
67.5
∘
=
1
2
2
+
2
{\displaystyle \sin {\frac {3\pi }{8}}=\sin 67.5^{\circ }={\tfrac {1}{2}}{\sqrt {2+{\sqrt {2}}}}\,}
cos
3
π
8
=
cos
67.5
∘
=
1
2
2
−
2
{\displaystyle \cos {\frac {3\pi }{8}}=\cos 67.5^{\circ }={\tfrac {1}{2}}{\sqrt {2-{\sqrt {2}}}}\,}
tan
3
π
8
=
tan
67.5
∘
=
2
+
1
{\displaystyle \tan {\frac {3\pi }{8}}=\tan 67.5^{\circ }={\sqrt {2}}+1\,}
cot
3
π
8
=
cot
67.5
∘
=
2
−
1
{\displaystyle \cot {\frac {3\pi }{8}}=\cot 67.5^{\circ }={\sqrt {2}}-1\,}
24°: sum 12° + 12°
sin
2
π
15
=
sin
24
∘
=
1
8
[
15
+
3
−
2
(
5
−
5
)
]
{\displaystyle \sin {\frac {2\pi }{15}}=\sin 24^{\circ }={\tfrac {1}{8}}\left[{\sqrt {15}}+{\sqrt {3}}-{\sqrt {2\left(5-{\sqrt {5}}\right)}}\right]\,}
cos
2
π
15
=
cos
24
∘
=
1
8
(
6
(
5
−
5
)
+
5
+
1
)
{\displaystyle \cos {\frac {2\pi }{15}}=\cos 24^{\circ }={\tfrac {1}{8}}\left({\sqrt {6\left(5-{\sqrt {5}}\right)}}+{\sqrt {5}}+1\right)\,}
tan
2
π
15
=
tan
24
∘
=
1
2
[
50
+
22
5
−
3
3
−
15
]
{\displaystyle \tan {\frac {2\pi }{15}}=\tan 24^{\circ }={\tfrac {1}{2}}\left[{\sqrt {50+22{\sqrt {5}}}}-3{\sqrt {3}}-{\sqrt {15}}\right]\,}
cot
2
π
15
=
cot
24
∘
=
1
2
[
15
−
3
+
2
(
5
−
5
)
]
{\displaystyle \cot {\frac {2\pi }{15}}=\cot 24^{\circ }={\tfrac {1}{2}}\left[{\sqrt {15}}-{\sqrt {3}}+{\sqrt {2\left(5-{\sqrt {5}}\right)}}\right]\,}
27°: sum 12° + 15°
sin
3
π
20
=
sin
27
∘
=
1
8
[
2
5
+
5
−
2
(
5
−
1
)
]
{\displaystyle \sin {\frac {3\pi }{20}}=\sin 27^{\circ }={\tfrac {1}{8}}\left[2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}\;\left({\sqrt {5}}-1\right)\right]\,}
cos
3
π
20
=
cos
27
∘
=
1
8
[
2
5
+
5
+
2
(
5
−
1
)
]
{\displaystyle \cos {\frac {3\pi }{20}}=\cos 27^{\circ }={\tfrac {1}{8}}\left[2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\;\left({\sqrt {5}}-1\right)\right]\,}
tan
3
π
20
=
tan
27
∘
=
5
−
1
−
5
−
2
5
{\displaystyle \tan {\frac {3\pi }{20}}=\tan 27^{\circ }={\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}\,}
cot
3
π
20
=
cot
27
∘
=
5
−
1
+
5
−
2
5
{\displaystyle \cot {\frac {3\pi }{20}}=\cot 27^{\circ }={\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}\,}
30°: regular hexagon
sin
π
6
=
sin
30
∘
=
1
2
{\displaystyle \sin {\frac {\pi }{6}}=\sin 30^{\circ }={\frac {1}{2}}\,}
cos
π
6
=
cos
30
∘
=
3
2
{\displaystyle \cos {\frac {\pi }{6}}=\cos 30^{\circ }={\frac {\sqrt {3}}{2}}\,}
tan
π
6
=
tan
30
∘
=
3
3
=
1
3
{\displaystyle \tan {\frac {\pi }{6}}=\tan 30^{\circ }={\frac {\sqrt {3}}{3}}={\frac {1}{\sqrt {3}}}\,}
cot
π
6
=
cot
30
∘
=
3
{\displaystyle \cot {\frac {\pi }{6}}=\cot 30^{\circ }={\sqrt {3}}\,}
60°: equilateral triangle
sin
π
3
=
sin
60
∘
=
3
2
{\displaystyle \sin {\frac {\pi }{3}}=\sin 60^{\circ }={\frac {\sqrt {3}}{2}}\,}
cos
π
3
=
cos
60
∘
=
1
2
{\displaystyle \cos {\frac {\pi }{3}}=\cos 60^{\circ }={\frac {1}{2}}\,}
tan
π
3
=
tan
60
∘
=
3
{\displaystyle \tan {\frac {\pi }{3}}=\tan 60^{\circ }={\sqrt {3}}\,}
cot
π
3
=
cot
60
∘
=
3
3
=
1
3
{\displaystyle \cot {\frac {\pi }{3}}=\cot 60^{\circ }={\frac {\sqrt {3}}{3}}={\frac {1}{\sqrt {3}}}\,}
33°: sum 15° + 18°
sin
11
π
60
=
sin
33
∘
=
1
16
[
2
(
3
−
1
)
5
+
5
+
2
(
1
+
3
)
(
5
−
1
)
]
{\displaystyle \sin {\frac {11\pi }{60}}=\sin 33^{\circ }={\tfrac {1}{16}}\left[2\left({\sqrt {3}}-1\right){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left(1+{\sqrt {3}}\right)\left({\sqrt {5}}-1\right)\right]\,}
cos
11
π
60
=
cos
33
∘
=
1
16
[
2
(
3
+
1
)
5
+
5
+
2
(
1
−
3
)
(
5
−
1
)
]
{\displaystyle \cos {\frac {11\pi }{60}}=\cos 33^{\circ }={\tfrac {1}{16}}\left[2\left({\sqrt {3}}+1\right){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left(1-{\sqrt {3}}\right)\left({\sqrt {5}}-1\right)\right]\,}
tan
11
π
60
=
tan
33
∘
=
1
4
[
2
−
(
2
−
3
)
(
3
+
5
)
]
[
2
+
2
(
5
−
5
)
]
{\displaystyle \tan {\frac {11\pi }{60}}=\tan 33^{\circ }={\tfrac {1}{4}}\left[2-\left(2-{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)\right]\left[2+{\sqrt {2\left(5-{\sqrt {5}}\right)}}\,\right]\,}
cot
11
π
60
=
cot
33
∘
=
1
4
[
2
−
(
2
+
3
)
(
3
+
5
)
]
[
2
−
2
(
5
−
5
)
]
{\displaystyle \cot {\frac {11\pi }{60}}=\cot 33^{\circ }={\tfrac {1}{4}}\left[2-\left(2+{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)\right]\left[2-{\sqrt {2\left(5-{\sqrt {5}}\right)}}\,\right]\,}
36°: regular pentagon [1]
sin
π
5
=
sin
36
∘
=
1
4
10
−
2
5
{\displaystyle \sin {\frac {\pi }{5}}=\sin 36^{\circ }={\frac {1}{4}}{\sqrt {10-2{\sqrt {5}}}}}
cos
π
5
=
cos
36
∘
=
5
+
1
4
=
φ
2
,
{\displaystyle \cos {\frac {\pi }{5}}=\cos 36^{\circ }={\frac {{\sqrt {5}}+1}{4}}={\frac {\varphi }{2}},}
φ golden ratio ;
tan
π
5
=
tan
36
∘
=
5
−
2
5
{\displaystyle \tan {\frac {\pi }{5}}=\tan 36^{\circ }={\sqrt {5-2{\sqrt {5}}}}\,}
cot
π
5
=
cot
36
∘
=
1
5
25
+
10
5
{\displaystyle \cot {\frac {\pi }{5}}=\cot 36^{\circ }={\frac {1}{5}}{\sqrt {25+10{\sqrt {5}}}}}
54°: sum 27° + 27°
sin
3
π
10
=
sin
54
∘
=
5
+
1
4
{\displaystyle \sin {\frac {3\pi }{10}}=\sin 54^{\circ }={\frac {{\sqrt {5}}+1}{4}}\,\!}
cos
3
π
10
=
cos
54
∘
=
10
−
2
5
4
{\displaystyle \cos {\frac {3\pi }{10}}=\cos 54^{\circ }={\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
tan
3
π
10
=
tan
54
∘
=
25
+
10
5
5
{\displaystyle \tan {\frac {3\pi }{10}}=\tan 54^{\circ }={\frac {\sqrt {25+10{\sqrt {5}}}}{5}}\,}
cot
3
π
10
=
cot
54
∘
=
5
−
2
5
{\displaystyle \cot {\frac {3\pi }{10}}=\cot 54^{\circ }={\sqrt {5-2{\sqrt {5}}}}\,}
39°: sum 18° + 21°
sin
13
π
60
=
sin
39
∘
=
1
16
[
2
(
1
−
3
)
5
−
5
+
2
(
3
+
1
)
(
5
+
1
)
]
{\displaystyle \sin {\frac {13\pi }{60}}=\sin 39^{\circ }={\tfrac {1}{16}}\left[2\left(1-{\sqrt {3}}\right){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {3}}+1\right)\left({\sqrt {5}}+1\right)\right]\,}
cos
13
π
60
=
cos
39
∘
=
1
16
[
2
(
1
+
3
)
5
−
5
+
2
(
3
−
1
)
(
5
+
1
)
]
{\displaystyle \cos {\frac {13\pi }{60}}=\cos 39^{\circ }={\tfrac {1}{16}}\left[2\left(1+{\sqrt {3}}\right){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {3}}-1\right)\left({\sqrt {5}}+1\right)\right]\,}
tan
13
π
60
=
tan
39
∘
=
1
4
[
(
2
−
3
)
(
3
−
5
)
−
2
]
[
2
−
2
(
5
+
5
)
]
{\displaystyle \tan {\frac {13\pi }{60}}=\tan 39^{\circ }={\tfrac {1}{4}}\left[\left(2-{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)-2\right]\left[2-{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}
cot
13
π
60
=
cot
39
∘
=
1
4
[
(
2
+
3
)
(
3
−
5
)
−
2
]
[
2
+
2
(
5
+
5
)
]
{\displaystyle \cot {\frac {13\pi }{60}}=\cot 39^{\circ }={\tfrac {1}{4}}\left[\left(2+{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)-2\right]\left[2+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}
42°: sum 21° + 21°
sin
7
π
30
=
sin
42
∘
=
30
+
6
5
−
5
+
1
8
{\displaystyle \sin {\frac {7\pi }{30}}=\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 {\frac {7\pi }{30}}=\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 {\frac {7\pi }{30}}=\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 {\frac {7\pi }{30}}=\cot 42^{\circ }={\frac {{\sqrt {50-22{\sqrt {5}}}}+3{\sqrt {3}}-{\sqrt {15}}}{2}}\,}
45°: square
sin
π
4
=
sin
45
∘
=
2
2
=
1
2
{\displaystyle \sin {\frac {\pi }{4}}=\sin 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}\,}
cos
π
4
=
cos
45
∘
=
2
2
=
1
2
{\displaystyle \cos {\frac {\pi }{4}}=\cos 45^{\circ }={\frac {\sqrt {2}}{2}}={\frac {1}{\sqrt {2}}}\,}
tan
π
4
=
tan
45
∘
=
1
{\displaystyle \tan {\frac {\pi }{4}}=\tan 45^{\circ }=1\,}
cot
π
4
=
cot
45
∘
=
1
{\displaystyle \cot {\frac {\pi }{4}}=\cot 45^{\circ }=1\,}
Baz
![{\displaystyle \tan \left({\frac {\pi }{60}}\right)=\tan \left(3^{\circ }\right)={\frac {\left[\left(2-{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)-2\right]\left[2-{\sqrt {10-2{\sqrt {5}}}}\right]}{4}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3def0c4623027bd006e5772927044dc5f2c82f41)
![{\displaystyle \cot \left({\frac {\pi }{60}}\right)=\cot \left(3^{\circ }\right)={\frac {\left[\left(2+{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)-2\right]\left[2+{\sqrt {10-2{\sqrt {5}}}}\right]}{4}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5604f62a7ebebb43eb6e64baa6f88872c7e9d069)
![{\displaystyle \sin {\frac {\pi }{15}}=\sin 12^{\circ }={\tfrac {1}{8}}\left[{\sqrt {2\left(5+{\sqrt {5}}\right)}}+{\sqrt {3}}-{\sqrt {15}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1228d7e22be483539e4ba91a0b4c9bfaf074eebe)
![{\displaystyle \cos {\frac {\pi }{15}}=\cos 12^{\circ }={\tfrac {1}{8}}\left[{\sqrt {6\left(5+{\sqrt {5}}\right)}}+{\sqrt {5}}-1\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2ac6239ea26bcfc4e7eefcc375f7f67b6a0acb)
![{\displaystyle \tan {\frac {\pi }{15}}=\tan 12^{\circ }={\tfrac {1}{2}}\left[3{\sqrt {3}}-{\sqrt {15}}-{\sqrt {2\left(25-11{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b57f4f65cc3f783c5b77b860a2297a8a8cbab4)
![{\displaystyle \cot {\frac {\pi }{15}}=\cot 12^{\circ }={\tfrac {1}{2}}\left[{\sqrt {15}}+{\sqrt {3}}+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe18dfa13f93c50f52d019bc3ea48fa0ca9bf472)
![{\displaystyle \sin {\frac {2\pi }{15}}=\sin 24^{\circ }={\tfrac {1}{8}}\left[{\sqrt {15}}+{\sqrt {3}}-{\sqrt {2\left(5-{\sqrt {5}}\right)}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c08024470722b423345f24249ea960c772fc87ae)
![{\displaystyle \tan {\frac {2\pi }{15}}=\tan 24^{\circ }={\tfrac {1}{2}}\left[{\sqrt {50+22{\sqrt {5}}}}-3{\sqrt {3}}-{\sqrt {15}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa351918ed8323175584ed4751bfacf5f61e67e)
![{\displaystyle \cot {\frac {2\pi }{15}}=\cot 24^{\circ }={\tfrac {1}{2}}\left[{\sqrt {15}}-{\sqrt {3}}+{\sqrt {2\left(5-{\sqrt {5}}\right)}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679f693d22aa397b2aa744452edc6d64496e09fd)
![{\displaystyle \sin {\frac {3\pi }{20}}=\sin 27^{\circ }={\tfrac {1}{8}}\left[2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}\;\left({\sqrt {5}}-1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/598d2e8e06f65d78a64d4c8450bff3de5e5c5220)
![{\displaystyle \cos {\frac {3\pi }{20}}=\cos 27^{\circ }={\tfrac {1}{8}}\left[2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\;\left({\sqrt {5}}-1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ed3130d2995af89ffd7fed92ca75b79e11b580)
![{\displaystyle \sin {\frac {11\pi }{60}}=\sin 33^{\circ }={\tfrac {1}{16}}\left[2\left({\sqrt {3}}-1\right){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left(1+{\sqrt {3}}\right)\left({\sqrt {5}}-1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67cd8c6ce776c45fc5b1dfaa972f24628f4d57ab)
![{\displaystyle \cos {\frac {11\pi }{60}}=\cos 33^{\circ }={\tfrac {1}{16}}\left[2\left({\sqrt {3}}+1\right){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left(1-{\sqrt {3}}\right)\left({\sqrt {5}}-1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89d8b1c326975404f0f829d7b88fcfd06abfee06)
![{\displaystyle \tan {\frac {11\pi }{60}}=\tan 33^{\circ }={\tfrac {1}{4}}\left[2-\left(2-{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)\right]\left[2+{\sqrt {2\left(5-{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb93b95e09248b579de718f6c6670c64b38a785)
![{\displaystyle \cot {\frac {11\pi }{60}}=\cot 33^{\circ }={\tfrac {1}{4}}\left[2-\left(2+{\sqrt {3}}\right)\left(3+{\sqrt {5}}\right)\right]\left[2-{\sqrt {2\left(5-{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/637555b8bd052954d2c6cff5ec40071d29d1ffeb)
![{\displaystyle \sin {\frac {13\pi }{60}}=\sin 39^{\circ }={\tfrac {1}{16}}\left[2\left(1-{\sqrt {3}}\right){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {3}}+1\right)\left({\sqrt {5}}+1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fde6336d3d6afde0136a7171d6f40a9e695c3436)
![{\displaystyle \cos {\frac {13\pi }{60}}=\cos 39^{\circ }={\tfrac {1}{16}}\left[2\left(1+{\sqrt {3}}\right){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {3}}-1\right)\left({\sqrt {5}}+1\right)\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d84363bc38d47561a10590df7ae08c6353619596)
![{\displaystyle \tan {\frac {13\pi }{60}}=\tan 39^{\circ }={\tfrac {1}{4}}\left[\left(2-{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)-2\right]\left[2-{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7739885ff94d55132c58be5a2ca59b17eb44037)
![{\displaystyle \cot {\frac {13\pi }{60}}=\cot 39^{\circ }={\tfrac {1}{4}}\left[\left(2+{\sqrt {3}}\right)\left(3-{\sqrt {5}}\right)-2\right]\left[2+{\sqrt {2\left(5+{\sqrt {5}}\right)}}\,\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a960ac308ab19d558f2f1c6399a1913bfe1f5ca)
