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1
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2
See also
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Template
:
Trigonometric Functions Exact Values Table
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Usage
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Calling
{{
Trigonometric Functions Exact Values Table
}}
will display:
Exact values of common angles
[1]
[2]
Radian
Degree
sin
cos
tan
cot
sec
csc
0
{\displaystyle 0}
0
∘
{\displaystyle 0^{\circ }}
0
{\displaystyle 0}
1
{\displaystyle 1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
1
{\displaystyle 1}
∞
{\displaystyle \infty }
π
24
{\displaystyle {\frac {\pi }{24}}}
7.5
∘
{\displaystyle 7.5^{\circ }}
1
2
2
−
2
+
3
{\displaystyle {\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}}
1
2
2
+
2
+
3
{\displaystyle {\frac {1}{2}}{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}
6
−
2
−
5
−
2
6
{\displaystyle {\sqrt {6}}-2-{\sqrt {5-2{\sqrt {6}}}}}
6
+
2
+
5
+
2
6
{\displaystyle {\sqrt {6}}+2+{\sqrt {5+2{\sqrt {6}}}}}
2
8
−
3
6
−
2
(
49
−
20
6
)
{\displaystyle {\sqrt {2}}{\sqrt {8-3{\sqrt {6}}-{\sqrt {2(49-20{\sqrt {6}})}}}}}
2
8
+
3
6
+
2
(
49
+
20
6
)
{\displaystyle {\sqrt {2}}{\sqrt {8+3{\sqrt {6}}+{\sqrt {2(49+20{\sqrt {6}})}}}}}
π
12
{\displaystyle {\frac {\pi }{12}}}
15
∘
{\displaystyle 15^{\circ }}
2
4
(
3
−
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
2
4
(
3
+
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
2
(
3
−
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}-1)}
2
(
3
+
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}+1)}
π
10
{\displaystyle {\frac {\pi }{10}}}
18
∘
{\displaystyle 18^{\circ }}
5
−
1
4
{\displaystyle {\frac {{\sqrt {5}}-1}{4}}}
10
+
2
5
4
{\displaystyle {\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
25
−
10
5
5
{\displaystyle {\frac {\sqrt {25-10{\sqrt {5}}}}{5}}}
5
+
2
5
{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}
50
−
10
5
5
{\displaystyle {\frac {\sqrt {50-10{\sqrt {5}}}}{5}}}
1
+
5
{\displaystyle 1+{\sqrt {5}}}
π
8
{\displaystyle {\frac {\pi }{8}}}
22.5
∘
{\displaystyle 22.5^{\circ }}
2
−
2
2
{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}
2
+
2
2
{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}
2
−
1
{\displaystyle {\sqrt {2}}-1}
2
+
1
{\displaystyle {\sqrt {2}}+1}
4
−
2
2
{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}
4
+
2
2
{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}
π
6
{\displaystyle {\frac {\pi }{6}}}
30
∘
{\displaystyle 30^{\circ }}
1
2
{\displaystyle {\frac {1}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
3
{\displaystyle {\sqrt {3}}}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
2
{\displaystyle 2}
π
5
{\displaystyle {\frac {\pi }{5}}}
36
∘
{\displaystyle 36^{\circ }}
10
−
2
5
4
{\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
1
+
5
4
{\displaystyle {\frac {1+{\sqrt {5}}}{4}}}
5
−
2
5
{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}
25
+
10
5
5
{\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{5}}}
5
−
1
{\displaystyle {\sqrt {5}}-1}
50
+
10
5
5
{\displaystyle {\frac {\sqrt {50+10{\sqrt {5}}}}{5}}}
π
4
{\displaystyle {\frac {\pi }{4}}}
45
∘
{\displaystyle 45^{\circ }}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
1
{\displaystyle 1}
1
{\displaystyle 1}
2
{\displaystyle {\sqrt {2}}}
2
{\displaystyle {\sqrt {2}}}
3
π
10
{\displaystyle {\frac {3\pi }{10}}}
54
∘
{\displaystyle 54^{\circ }}
1
+
5
4
{\displaystyle {\frac {1+{\sqrt {5}}}{4}}}
10
−
2
5
4
{\displaystyle {\frac {\sqrt {10-2{\sqrt {5}}}}{4}}}
25
+
10
5
5
{\displaystyle {\frac {\sqrt {25+10{\sqrt {5}}}}{5}}}
5
−
2
5
{\displaystyle {\sqrt {5-2{\sqrt {5}}}}}
50
+
10
5
5
{\displaystyle {\frac {\sqrt {50+10{\sqrt {5}}}}{5}}}
5
−
1
{\displaystyle {\sqrt {5}}-1}
π
3
{\displaystyle {\frac {\pi }{3}}}
60
∘
{\displaystyle 60^{\circ }}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
3
{\displaystyle {\sqrt {3}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
2
{\displaystyle 2}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
3
π
8
{\displaystyle {\frac {3\pi }{8}}}
67.5
∘
{\displaystyle 67.5^{\circ }}
2
+
2
2
{\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}
2
−
2
2
{\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}
2
+
1
{\displaystyle {\sqrt {2}}+1}
2
−
1
{\displaystyle {\sqrt {2}}-1}
4
+
2
2
{\displaystyle {\sqrt {4+2{\sqrt {2}}}}}
4
−
2
2
{\displaystyle {\sqrt {4-2{\sqrt {2}}}}}
2
π
5
{\displaystyle {\frac {2\pi }{5}}}
72
∘
{\displaystyle 72^{\circ }}
10
+
2
5
4
{\displaystyle {\frac {\sqrt {10+2{\sqrt {5}}}}{4}}}
5
−
1
4
{\displaystyle {\frac {{\sqrt {5}}-1}{4}}}
5
+
2
5
{\displaystyle {\sqrt {5+2{\sqrt {5}}}}}
25
−
10
5
5
{\displaystyle {\frac {\sqrt {25-10{\sqrt {5}}}}{5}}}
1
+
5
{\displaystyle 1+{\sqrt {5}}}
50
−
10
5
5
{\displaystyle {\frac {\sqrt {50-10{\sqrt {5}}}}{5}}}
5
π
12
{\displaystyle {\frac {5\pi }{12}}}
75
∘
{\displaystyle 75^{\circ }}
2
4
(
3
+
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
2
4
(
3
−
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
2
(
3
+
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}+1)}
2
(
3
−
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}-1)}
π
2
{\displaystyle {\frac {\pi }{2}}}
90
∘
{\displaystyle 90^{\circ }}
1
{\displaystyle 1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
0
{\displaystyle 0}
∞
{\displaystyle \infty }
1
{\displaystyle 1}
7
π
12
{\displaystyle {\frac {7\pi }{12}}}
105
∘
{\displaystyle 105^{\circ }}
2
4
(
3
+
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
−
2
4
(
3
−
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
−
2
−
3
{\displaystyle -2-{\sqrt {3}}}
−
2
+
3
{\displaystyle -2+{\sqrt {3}}}
−
2
(
1
+
3
)
{\displaystyle -{\sqrt {2}}(1+{\sqrt {3}})}
2
(
3
−
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}-1)}
2
π
3
{\displaystyle {\frac {2\pi }{3}}}
120
∘
{\displaystyle 120^{\circ }}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
−
1
2
{\displaystyle -{\frac {1}{2}}}
−
3
{\displaystyle -{\sqrt {3}}}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
−
2
{\displaystyle -2}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
3
π
4
{\displaystyle {\frac {3\pi }{4}}}
135
∘
{\displaystyle 135^{\circ }}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
−
2
2
{\displaystyle -{\frac {\sqrt {2}}{2}}}
−
1
{\displaystyle -1}
−
1
{\displaystyle -1}
−
2
{\displaystyle -{\sqrt {2}}}
2
{\displaystyle {\sqrt {2}}}
5
π
6
{\displaystyle {\frac {5\pi }{6}}}
150
∘
{\displaystyle 150^{\circ }}
1
2
{\displaystyle {\frac {1}{2}}}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
−
3
{\displaystyle -{\sqrt {3}}}
−
2
3
3
{\displaystyle -{\frac {2{\sqrt {3}}}{3}}}
2
{\displaystyle 2}
11
π
12
{\displaystyle {\frac {11\pi }{12}}}
165
∘
{\displaystyle 165^{\circ }}
2
4
(
3
−
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
−
2
4
(
3
+
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
−
2
−
3
{\displaystyle -2-{\sqrt {3}}}
−
2
+
3
{\displaystyle -2+{\sqrt {3}}}
−
2
(
3
−
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}-1)}
2
(
3
+
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}+1)}
π
{\displaystyle \pi }
180
∘
{\displaystyle 180^{\circ }}
0
{\displaystyle 0}
−
1
{\displaystyle -1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
−
1
{\displaystyle -1}
∞
{\displaystyle \infty }
13
π
12
{\displaystyle {\frac {13\pi }{12}}}
195
∘
{\displaystyle 195^{\circ }}
−
3
−
1
2
2
{\displaystyle -{\frac {{\sqrt {3}}-1}{2{\sqrt {2}}}}}
−
3
+
1
2
2
{\displaystyle -{\frac {{\sqrt {3}}+1}{2{\sqrt {2}}}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
−
2
(
3
−
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}-1)}
−
2
(
1
+
3
)
{\displaystyle -{\sqrt {2}}(1+{\sqrt {3}})}
7
π
6
{\displaystyle {\frac {7\pi }{6}}}
210
∘
{\displaystyle 210^{\circ }}
−
1
2
{\displaystyle -{\frac {1}{2}}}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
3
{\displaystyle {\sqrt {3}}}
−
2
3
3
{\displaystyle -{\frac {2{\sqrt {3}}}{3}}}
−
2
{\displaystyle -2}
5
π
4
{\displaystyle {\frac {5\pi }{4}}}
225
∘
{\displaystyle 225^{\circ }}
−
2
2
{\displaystyle -{\dfrac {\sqrt {2}}{2}}}
−
2
2
{\displaystyle -{\dfrac {\sqrt {2}}{2}}}
1
{\displaystyle 1}
1
{\displaystyle 1}
−
2
{\displaystyle -{\sqrt {2}}}
−
2
{\displaystyle -{\sqrt {2}}}
4
π
3
{\displaystyle {\frac {4\pi }{3}}}
240
∘
{\displaystyle 240^{\circ }}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
1
2
{\displaystyle -{\frac {1}{2}}}
3
{\displaystyle {\sqrt {3}}}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
−
2
{\displaystyle -2}
−
2
3
3
{\displaystyle -{\frac {2{\sqrt {3}}}{3}}}
17
π
12
{\displaystyle {\frac {17\pi }{12}}}
255
∘
{\displaystyle 255^{\circ }}
−
2
4
(
3
+
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
−
2
4
(
3
−
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
2
+
3
{\displaystyle 2+{\sqrt {3}}}
2
−
3
{\displaystyle 2-{\sqrt {3}}}
−
2
(
3
+
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}+1)}
−
2
(
3
−
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}-1)}
3
π
2
{\displaystyle {\frac {3\pi }{2}}}
270
∘
{\displaystyle 270^{\circ }}
−
1
{\displaystyle -1}
0
{\displaystyle 0}
∞
{\displaystyle \infty }
0
{\displaystyle 0}
∞
{\displaystyle \infty }
−
1
{\displaystyle -1}
19
π
12
{\displaystyle {\frac {19\pi }{12}}}
285
∘
{\displaystyle 285^{\circ }}
−
2
4
(
3
+
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
2
4
(
3
−
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
−
2
+
3
{\displaystyle -2+{\sqrt {3}}}
−
2
+
3
{\displaystyle -2+{\sqrt {3}}}
2
(
3
+
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}+1)}
−
2
(
3
−
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}-1)}
5
π
3
{\displaystyle {\frac {5\pi }{3}}}
300
∘
{\displaystyle 300^{\circ }}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
−
3
{\displaystyle -{\sqrt {3}}}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
2
{\displaystyle 2}
−
2
3
3
{\displaystyle -{\frac {2{\sqrt {3}}}{3}}}
7
π
4
{\displaystyle {\frac {7\pi }{4}}}
315
∘
{\displaystyle 315^{\circ }}
−
2
2
{\displaystyle -{\frac {\sqrt {2}}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
−
1
{\displaystyle -1}
−
1
{\displaystyle -1}
2
{\displaystyle {\sqrt {2}}}
−
2
{\displaystyle -{\sqrt {2}}}
11
π
6
{\displaystyle {\frac {11\pi }{6}}}
330
∘
{\displaystyle 330^{\circ }}
−
1
2
{\displaystyle -{\frac {1}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
−
3
{\displaystyle -{\sqrt {3}}}
2
3
3
{\displaystyle {\frac {2{\sqrt {3}}}{3}}}
−
2
{\displaystyle -2}
23
π
12
{\displaystyle {\frac {23\pi }{12}}}
345
∘
{\displaystyle 345^{\circ }}
−
2
4
(
3
−
1
)
{\displaystyle -{\frac {\sqrt {2}}{4}}({\sqrt {3}}-1)}
2
4
(
3
+
1
)
{\displaystyle {\frac {\sqrt {2}}{4}}({\sqrt {3}}+1)}
−
2
+
3
{\displaystyle -2+{\sqrt {3}}}
−
2
−
3
{\displaystyle -2-{\sqrt {3}}}
2
(
3
−
1
)
{\displaystyle {\sqrt {2}}({\sqrt {3}}-1)}
−
2
(
3
+
1
)
{\displaystyle -{\sqrt {2}}({\sqrt {3}}+1)}
See also
[
edit
]
Exact trigonometric values
List of trigonometric identities
Template:DomainsImagesAndPrototypesOfTrigAndInverseTrigFunctions
Template:Trigonometry
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^
Abramowitz & Stegun 1972
, p. 74, 4.3.46
^
Surgent, Scott (November 2018).
"Exact Values of Sine and Cosine of Angles in Increments of 3 Degrees"
(PDF)
.
Scott Surgent's ASU Website
. Wayback Machine.
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