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# Valeurs trigonométriques exactes

Bibliothèque wikiversitaire
Intitulé : Valeurs trigonométriques exactes

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