Valeurs trigonométriques exactes

Bibliothèque wikiversitaire

Intitulé : Valeurs trigonométriques exactes

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Angle en radian	Angle en degré	Cosinus	Sinus	Tangente
$0$	0°	$1$	$0$	$0$
${\frac {\pi }{60}}$	3°	${\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}$	${\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)-2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}}$	${\frac {\left((2-{\sqrt {3}})(3+{\sqrt {5}})-2\right)\left(2-{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}\right)}{4}}$
${\frac {\pi }{32}}$	5,625°	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}\left({\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}-1\right)-{\sqrt {2}}-1$
${\frac {\pi }{30}}$	6°	${\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {3}}+{\sqrt {15}}}{8}}$	${\frac {{\sqrt {30-6{\sqrt {5}}}}-{\sqrt {5}}-1}{8}}$	${\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}-{\sqrt {3}}({\sqrt {5}}-1)}{2}}$
${\frac {\pi }{24}}$	7,5°	${\sqrt {\frac {4+{\sqrt {2}}+{\sqrt {6}}}{8}}}$	${\sqrt {\frac {4-{\sqrt {2}}-{\sqrt {6}}}{8}}}$	${\sqrt {6}}-{\sqrt {3}}+{\sqrt {2}}-2$
${\frac {\pi }{20}}$	9°	${\frac {{\sqrt {2}}\left({\sqrt {5}}+1\right)+2{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {2}}\left({\sqrt {5}}+1\right)-2{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}$
${\frac {\pi }{16}}$	11,25°	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2}}-1$
${\frac {\pi }{15}}$	12°	${\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {5}}-1}{8}}$	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}-{\sqrt {3}}({\sqrt {5}}-1)}{8}}$	${\frac {{\sqrt {3}}(3-{\sqrt {5}})-{\sqrt {2}}{\sqrt {25-11{\sqrt {5}}}}}{2}}$
${\frac {\pi }{12}}$	15°	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	$2-{\sqrt {3}}$
${\frac {3\pi }{32}}$	16,875°	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}\left({\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}-1\right)-{\sqrt {2}}+1$
${\frac {\pi }{10}}$	18°	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{4}}$	${\frac {{\sqrt {5}}-1}{4}}$	${\frac {{\sqrt {5}}{\sqrt {5-2{\sqrt {5}}}}}{5}}$
${\frac {7\pi }{60}}$	21°	${\frac {2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)}{16}}$	${\frac {2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)}{16}}$	${\frac {\left(2-(2+{\sqrt {3}})(3-{\sqrt {5}})\right)\left(2-{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)}{4}}$
${\frac {\pi }{8}}$	22,5°	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\sqrt {2}}-1$
${\frac {2\pi }{15}}$	24°	${\frac {{\sqrt {5}}+1+{\sqrt {6}}{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {2}}{\sqrt {25+11{\sqrt {5}}}}-{\sqrt {3}}(3+{\sqrt {5}})}{2}}$
${\frac {3\pi }{20}}$	27°	${\frac {2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {5}}-1\right)}{8}}$	${\frac {2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}\left({\sqrt {5}}-1\right)}{8}}$	${\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}$
${\frac {5\pi }{32}}$	28,125°	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}\left({\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}-1\right)+{\sqrt {2}}-1$
${\frac {\pi }{6}}$	30°	${\frac {\sqrt {3}}{2}}$	${\frac {1}{2}}$	${\frac {\sqrt {3}}{3}}$
${\frac {11\pi }{60}}$	33°	${\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}$	${\frac {2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)}{16}}$	${\frac {\left(2-(2-{\sqrt {3}})(3+{\sqrt {5}})\right)\left(2+{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}\right)}{4}}$
${\frac {3\pi }{16}}$	33,75°	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}-{\sqrt {2}}+1$
${\frac {\pi }{5}}$	36°	${\frac {{\sqrt {5}}+1}{4}}$	${\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{4}}$	${\sqrt {5-2{\sqrt {5}}}}$
${\frac {5\pi }{24}}$	37,5°	${\sqrt {\frac {4-{\sqrt {2}}+{\sqrt {6}}}{8}}}$	${\sqrt {\frac {4+{\sqrt {2}}-{\sqrt {6}}}{8}}}$	${\sqrt {6}}+{\sqrt {3}}-{\sqrt {2}}-2$
${\frac {7\pi }{30}}$	39°	${\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}}$	${\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)-2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}}$	${\frac {\left((2-{\sqrt {3}})(3-{\sqrt {5}})-2\right)\left(2-{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)}{4}}$
${\frac {7\pi }{32}}$	39,375°	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}\left({\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}-1\right)+{\sqrt {2}}+1$
${\frac {7\pi }{30}}$	42°	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {3}}({\sqrt {5}}-1)}{8}}$	${\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}-({\sqrt {5}}-1)}{8}}$	${\frac {{\sqrt {3}}\left({\sqrt {5}}+1\right)-{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{2}}$
${\frac {\pi }{4}}$	45°	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {2}}{2}}$	$1$
${\frac {4\pi }{15}}$	48°	${\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}-({\sqrt {5}}-1)}{8}}$	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {3}}({\sqrt {5}}-1)}{8}}$	${\frac {\left((2+{\sqrt {3}})(3+{\sqrt {5}})-2\right)\left(2+{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}\right)}{4}}$
${\frac {9\pi }{32}}$	50,625°	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}\left({\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}+1\right)-{\sqrt {2}}-1$
${\frac {17\pi }{60}}$	51°	${\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)-2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}}$	${\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}}$	${\frac {\left((2+{\sqrt {3}})(3-{\sqrt {5}})-2\right)\left(2+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)}{4}}$
${\frac {7\pi }{24}}$	52,5°	${\sqrt {\frac {4+{\sqrt {2}}-{\sqrt {6}}}{8}}}$	${\sqrt {\frac {4-{\sqrt {2}}+{\sqrt {6}}}{8}}}$	${\sqrt {6}}-{\sqrt {3}}-{\sqrt {2}}+2$
${\frac {3\pi }{10}}$	54°	${\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{4}}$	${\frac {{\sqrt {5}}+1}{4}}$	${\frac {{\sqrt {5}}{\sqrt {5+2{\sqrt {5}}}}}{5}}$
${\frac {5\pi }{16}}$	56,25°	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}+{\sqrt {2}}-1$
${\frac {19\pi }{60}}$	57°	${\frac {2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)}{16}}$	${\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}$	${\frac {\left(2-(2+{\sqrt {3}})(3+{\sqrt {5}})\right)\left(2-{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}\right)}{4}}$
${\frac {\pi }{3}}$	60°	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$	${\sqrt {3}}$
${\frac {11\pi }{32}}$	61,875°	${\frac {\sqrt {2-{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}\left({\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}+1\right)-{\sqrt {2}}+1$
${\frac {7\pi }{20}}$	63°	${\frac {2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}\left({\sqrt {5}}-1\right)}{8}}$	${\frac {2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}\left({\sqrt {5}}-1\right)}{8}}$	${\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}$
${\frac {11\pi }{30}}$	66°	${\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {5}}+1+{\sqrt {6}}{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {3}}({\sqrt {5}}-1)+{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{2}}$
${\frac {3\pi }{8}}$	67,5°	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$	${\sqrt {2}}+1$
${\frac {23\pi }{60}}$	69°	${\frac {2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)}{16}}$	${\frac {2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)}{16}}$	${\frac {\left(2-(2-{\sqrt {3}})(3-{\sqrt {5}})\right)\left(2+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}\right)}{4}}$
${\frac {2\pi }{5}}$	72°	${\frac {{\sqrt {5}}-1}{4}}$	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{4}}$	${\sqrt {5+2{\sqrt {5}}}}$
${\frac {13\pi }{32}}$	73,175°	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4-2{\sqrt {2}}}}\left({\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}+1\right)+{\sqrt {2}}-1$
${\frac {5\pi }{12}}$	75°	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$	$2+{\sqrt {3}}$
${\frac {13\pi }{30}}$	78°	${\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}-{\sqrt {3}}({\sqrt {5}}-1)}{8}}$	${\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {5}}-1}{8}}$	${\frac {{\sqrt {3}}({\sqrt {5}}+1)+{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{2}}$
${\frac {7\pi }{16}}$	78,75°	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}+{\sqrt {2}}+1$
${\frac {9\pi }{20}}$	81°	${\frac {{\sqrt {2}}\left({\sqrt {5}}+1\right)-2{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\frac {{\sqrt {2}}\left({\sqrt {5}}+1\right)+2{\sqrt {5-{\sqrt {5}}}}}{8}}$	${\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}$
${\frac {11\pi }{24}}$	82,5°	${\sqrt {\frac {4-{\sqrt {2}}-{\sqrt {6}}}{8}}}$	${\sqrt {\frac {4+{\sqrt {2}}+{\sqrt {6}}}{8}}}$	${\sqrt {6}}+{\sqrt {3}}+{\sqrt {2}}+2$
${\frac {7\pi }{15}}$	84°	${\frac {{\sqrt {30-6{\sqrt {5}}}}-{\sqrt {5}}-1}{8}}$	${\frac {{\sqrt {10-2{\sqrt {5}}}}+{\sqrt {3}}+{\sqrt {15}}}{8}}$	${\frac {3{\sqrt {3}}+{\sqrt {15}}+{\sqrt {50+22{\sqrt {5}}}}}{2}}$
${\frac {15\pi }{32}}$	84,375°	${\frac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$	${\sqrt {4+2{\sqrt {2}}}}\left({\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+1\right)+{\sqrt {2}}+1$
${\frac {29\pi }{60}}$	87°	${\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)-2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}}$	${\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}$	${\frac {\left((2+{\sqrt {3}})(3+{\sqrt {5}})-2\right)\left(2+{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}\right)}{4}}$
${\frac {\pi }{2}}$	90°	$0$	$1$	non définie

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