# Valeurs trigonométriques exactes

| 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 |