User:LRT505/三角函數精確值 (非整數角度)

${\displaystyle \mathbf {0} }$ ${\displaystyle {\tfrac {1}{12}}}$ ${\displaystyle {\tfrac {1}{8}}}$ ${\displaystyle {\tfrac {1}{6}}}$ ${\displaystyle {\tfrac {1}{4}}}$ ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle {\tfrac {3}{4}}}$ ${\displaystyle \mathbf {1} }$

3°/16

${\displaystyle {}_{\sin {\frac {\pi }{960}}=\sin 0.1875^{\circ }={\frac {\sqrt {8-{\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+2{\sqrt {3}}+2{\sqrt {15}}+2{\sqrt {10-2{\sqrt {5}}}}}}}}}}}}}}{4}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{960}}=\cos 0.1875^{\circ }={\frac {\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+2{\sqrt {3}}+2{\sqrt {15}}+2{\sqrt {10-2{\sqrt {5}}}}}}}}}}}}}}{4}}}\,}$

2.5°

${\displaystyle {}_{\sin {\frac {\pi }{72}}=\sin 2.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{72}}=\cos 2.5^{\circ }={\tfrac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}$

${\displaystyle {}_{\tan {\frac {\pi }{72}}=\tan 2.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}-{\frac {1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}-{\frac {1-{\sqrt {3}}{\mathrm {i} }}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}}\,}$

3.75°

${\displaystyle {}_{\sin {\frac {\pi }{48}}=\sin 3.75^{\circ }={\tfrac {{\sqrt {6+3{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12+6{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12+6{\sqrt {2+{\sqrt {2}}}}+6{\sqrt {2}}+3{\sqrt {4+2{\sqrt {2}}}}}}-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {2}}-{\sqrt {4+2{\sqrt {2}}}}}}}{4}}={\tfrac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{48}}=\cos 3.75^{\circ }={\tfrac {{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}}+{\sqrt {6-3{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12-6{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12-6{\sqrt {2+{\sqrt {2}}}}+6{\sqrt {2}}-3{\sqrt {4+2{\sqrt {2}}}}}}}{4}}={\tfrac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}\,}$

4.5°

${\displaystyle {}_{\sin {\frac {\pi }{40}}=\sin 4.5^{\circ }={\tfrac {2{\sqrt {10+2{\sqrt {5}}+5{\sqrt {2}}+{\sqrt {1}}0}}-{\sqrt {20+4{\sqrt {5}}+10{\sqrt {2}}+2{\sqrt {10}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {20-10{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{40}}=\cos 4.5^{\circ }={\tfrac {2{\sqrt {10+2{\sqrt {5}}-5{\sqrt {2}}-{\sqrt {1}}0}}+{\sqrt {20+4{\sqrt {5}}-10{\sqrt {2}}-2{\sqrt {10}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}}}}{8}}}\,}$

7.5°

${\displaystyle {}_{\sin {\frac {\pi }{24}}=\sin 7.5^{\circ }={\frac {{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {12+6{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{24}}=\cos 7.5^{\circ }={\tfrac {{\sqrt {6-3{\sqrt {2}}}}+{\sqrt {12-6{\sqrt {2}}}}+{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\tan {\frac {\pi }{24}}=\tan 7.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}}\,}$

${\displaystyle {}_{\cot {\frac {\pi }{24}}=\cot 7.5^{\circ }=2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}}}\,}$

${\displaystyle {}_{\csc {\frac {\pi }{24}}=\csc 7.5^{\circ }=4{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {24-6{\sqrt {2}}-6{\sqrt {6}}}}+5{\sqrt {4-{\sqrt {2}}-{\sqrt {6}}}}+3{\sqrt {12-3{\sqrt {2}}-3{\sqrt {6}}}}}\,}$

${\displaystyle {}_{\sec {\frac {\pi }{24}}=\sec 7.5^{\circ }=4{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {24+6{\sqrt {2}}+6{\sqrt {6}}}}-5{\sqrt {4+{\sqrt {2}}+{\sqrt {6}}}}-3{\sqrt {12+3{\sqrt {2}}+3{\sqrt {6}}}}}\,}$

11.25°

${\displaystyle {}_{\sin {\frac {\pi }{16}}=\sin 11.25^{\circ }={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}{2}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{16}}=\cos 11.25^{\circ }={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}}\,}$

${\displaystyle {}_{\tan {\frac {\pi }{16}}=\tan 11.25^{\circ }={\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2}}-1}\,}$

${\displaystyle {}_{\cot {\frac {\pi }{16}}=\cot 11.25^{\circ }={\sqrt {4+2{\sqrt {2}}}}+{\sqrt {2}}+1}\,}$

${\displaystyle {}_{\csc {\frac {\pi }{16}}=\csc 11.25^{\circ }=4{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4+2{\sqrt {2}}-2{\sqrt {2+{\sqrt {2}}}}-{\sqrt {4+2{\sqrt {2}}}}}}+{\sqrt {8+4{\sqrt {2}}-4{\sqrt {2+{\sqrt {2}}}}-2{\sqrt {4+2{\sqrt {2}}}}}}}\,}$

${\displaystyle {}_{\sec {\frac {\pi }{16}}=\sec 11.25^{\circ }=4{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}}}-2{\sqrt {4+2{\sqrt {2}}+2{\sqrt {2+{\sqrt {2}}}}+{\sqrt {4+2{\sqrt {2}}}}}}-{\sqrt {8+4{\sqrt {2}}+4{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {4+2{\sqrt {2}}}}}}}\,}$

12.5°

${\displaystyle {}_{\sin {\frac {5\pi }{72}}=\sin 12.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {5\pi }{72}}=\cos 12.5^{\circ }={\tfrac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\mathrm {i} }}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\mathrm {i} }}}}{4}}}\,}$

13.5°

${\displaystyle {}_{\sin {\frac {3\pi }{40}}=\sin 13.5^{\circ }={\frac {{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {3\pi }{40}}=\cos 13.5^{\circ }={\frac {{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}$

17.5°

${\displaystyle {}_{\sin {\frac {7\pi }{72}}=\sin 17.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {7\pi }{72}}=\cos 17.5^{\circ }={\frac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}$

22.5°

${\displaystyle {}_{\sin {\frac {\pi }{8}}=\sin 22.5^{\circ }={\tfrac {\sqrt {2-{\sqrt {2}}}}{2}}}\,}$

${\displaystyle {}_{\cos {\frac {\pi }{8}}=\cos 22.5^{\circ }={\tfrac {\sqrt {2+{\sqrt {2}}}}{2}}}\,}$

${\displaystyle {}_{\tan {\frac {\pi }{8}}=\tan 22.5^{\circ }={\sqrt {2}}-1}\,}$

${\displaystyle {}_{\cot {\frac {\pi }{8}}=\cot 22.5^{\circ }={\sqrt {2}}+1}\,}$

${\displaystyle {}_{\csc {\frac {\pi }{8}}=\csc 22.5^{\circ }={\sqrt {4+2{\sqrt {2}}}}}\,}$

${\displaystyle {}_{\sec {\frac {\pi }{8}}=\sec 22.5^{\circ }={\sqrt {4-2{\sqrt {2}}}}}\,}$

27.5°

${\displaystyle {}_{\sin {\frac {11\pi }{72}}=\sin 27.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {11\pi }{72}}=\cos 27.5^{\circ }={\frac {{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}$

${\displaystyle {}_{\cot {\frac {11\pi }{72}}=\cot 27.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}+{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}+{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}}\,}$

31.5°

${\displaystyle {}_{\sin {\frac {7\pi }{40}}=\sin 31.5^{\circ }={\frac {{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {7\pi }{40}}=\cos 31.5^{\circ }={\frac {{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}}{8}}}\,}$

32.5°

${\displaystyle {}_{\sin {\frac {13\pi }{72}}=\sin 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {13\pi }{72}}=\cos 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cot {\frac {13\pi }{72}}=\cot 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}-{\sqrt {6}}-{\sqrt {2}}+2+{\sqrt {3}}}\,}$

33.75°

${\displaystyle {}_{\sin {\frac {3\pi }{16}}=\sin 33.75^{\circ }={\tfrac {\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}{2}}}\,}$

${\displaystyle {}_{\cos {\frac {3\pi }{16}}=\cos 33.75^{\circ }={\tfrac {\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}{2}}}\,}$

${\displaystyle {}_{\tan {\frac {3\pi }{16}}=\tan 33.75^{\circ }={\sqrt {4-2{\sqrt {2}}}}-{\sqrt {2}}+1}\,}$

${\displaystyle {}_{\cot {\frac {3\pi }{16}}=\cot 33.75^{\circ }={\sqrt {4-2{\sqrt {2}}}}+{\sqrt {2}}-1}\,}$

${\displaystyle {}_{\csc {\frac {3\pi }{16}}=\csc 33.75^{\circ }=4{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4-2{\sqrt {2-{\sqrt {2}}}}}}+2{\sqrt {4-2{\sqrt {2}}-2{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}}}-{\sqrt {8-4{\sqrt {2}}-4{\sqrt {2-{\sqrt {2}}}}+2{\sqrt {4-2{\sqrt {2}}}}}}}\,}$

${\displaystyle {}_{\sec {\frac {3\pi }{16}}=\sec 33.75^{\circ }=4{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4+2{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4-2{\sqrt {2}}+2{\sqrt {2-{\sqrt {2}}}}-{\sqrt {4-2{\sqrt {2}}}}}}+{\sqrt {8-4{\sqrt {2}}+4{\sqrt {2-{\sqrt {2}}}}-2{\sqrt {4-2{\sqrt {2}}}}}}}\,}$

37.5°

${\displaystyle {}_{\sin {\frac {5\pi }{24}}=\sin 37.5^{\circ }={\tfrac {{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\cos {\frac {5\pi }{24}}=\cos 37.5^{\circ }={\tfrac {{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\tan {\frac {5\pi }{24}}=\tan 37.5^{\circ }={\sqrt {3}}+{\sqrt {6}}-{\sqrt {2}}-2}\,}$

${\displaystyle {}_{\cot {\frac {5\pi }{24}}=\cot 37.5^{\circ }=2+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}\,}$

${\displaystyle {}_{\csc {\frac {5\pi }{24}}=\csc 37.5^{\circ }=4{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {24+6{\sqrt {2}}-6{\sqrt {6}}}}-5{\sqrt {4+{\sqrt {2}}-{\sqrt {6}}}}+3{\sqrt {12+3{\sqrt {2}}-3{\sqrt {6}}}}}\,}$

${\displaystyle {}_{\sec {\frac {5\pi }{24}}=\sec 37.5^{\circ }=4{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {24-6{\sqrt {2}}+6{\sqrt {6}}}}+5{\sqrt {4+{\sqrt {6}}-{\sqrt {2}}}}+3{\sqrt {12+3{\sqrt {6}}-3{\sqrt {2}}}}}\,}$

40.5°

${\displaystyle {}_{\sin {\frac {9\pi }{40}}=\sin 40.5^{\circ }={\frac {{\sqrt {20-2{\sqrt {10}}+4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {9\pi }{40}}=\cos 40.5^{\circ }={\frac {{\sqrt {20+2{\sqrt {10}}+4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\tan {\frac {9\pi }{40}}=\tan 40.5^{\circ }={\frac {3{\sqrt {18-2{\sqrt {5}}+8{\sqrt {5-{\sqrt {5}}}}}}-2{\sqrt {45-5{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}}}+{\sqrt {126+20{\sqrt {2}}+56{\sqrt {5-{\sqrt {5}}}}-14{\sqrt {5}}-36{\sqrt {10}}-16{\sqrt {50-10{\sqrt {5}}}}}}}{4}}}\,}$

${\displaystyle {}_{\cot {\frac {9\pi }{40}}=\cot 40.5^{\circ }={\frac {13{\sqrt {18-2{\sqrt {5}}+8{\sqrt {5-{\sqrt {5}}}}}}+7{\sqrt {90-10{\sqrt {5}}+40{\sqrt {5-{\sqrt {5}}}}}}-16{\sqrt {25-7{\sqrt {5}}+10{\sqrt {5-{\sqrt {5}}}}-2{\sqrt {25-5{\sqrt {5}}}}}}+22{\sqrt {9-{\sqrt {5}}+4{\sqrt {5-{\sqrt {5}}}}}}+6{\sqrt {45-5{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}}}-12{\sqrt {50-14{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}-4{\sqrt {25-5{\sqrt {5}}}}}}-8{\sqrt {125-35{\sqrt {5}}+50{\sqrt {5-{\sqrt {5}}}}-10{\sqrt {25-5{\sqrt {5}}}}}}-4{\sqrt {250-70{\sqrt {5}}+100{\sqrt {5-{\sqrt {5}}}}-20{\sqrt {25-5{\sqrt {5}}}}}}}{4}}}\,}$

${\displaystyle {}_{={\frac {{\sqrt {22+6{\sqrt {5}}-8{\sqrt {10+4{\sqrt {5}}}}}}+4{\sqrt {11+3{\sqrt {5}}-4{\sqrt {10+4{\sqrt {5}}}}}}+{\sqrt {110+30{\sqrt {5}}-40{\sqrt {10+4{\sqrt {5}}}}}}}{4}}}\,}$

${\displaystyle {}_{\sec {\frac {9\pi }{40}}=\sec 40.5^{\circ }={\frac {4{\sqrt {10+2{\sqrt {5}}-5{\sqrt {2}}-{\sqrt {10}}}}+{\sqrt {20+4{\sqrt {5}}-10{\sqrt {2}}-2{\sqrt {10}}}}-{\sqrt {100+20{\sqrt {5}}-50{\sqrt {2}}-10{\sqrt {10}}}}+2{\sqrt {20+10{\sqrt {2}}}}+6{\sqrt {4+2{\sqrt {2}}}}-2{\sqrt {10+5{\sqrt {2}}}}-10{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}$

42.5°

${\displaystyle {}_{\sin {\frac {17\pi }{72}}=\sin 42.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {17\pi }{72}}=\cos 42.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}$

49.5°

${\displaystyle {}_{\sin {\frac {11\pi }{40}}=\sin 49.5^{\circ }={\tfrac {{\sqrt {20+2{\sqrt {10}}+4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {11\pi }{40}}=\cos 49.5^{\circ }={\tfrac {{\sqrt {20-2{\sqrt {10}}+4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}$

58.5°

${\displaystyle {}_{\sin {\frac {13\pi }{40}}=\sin 58.5^{\circ }={\tfrac {{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {13\pi }{40}}=\cos 58.5^{\circ }={\tfrac {{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}}{8}}}\,}$

67.5°

${\displaystyle {}_{\sin {\frac {3\pi }{8}}=\sin 67.5^{\circ }={\tfrac {\sqrt {2+{\sqrt {2}}}}{2}}}\,}$

${\displaystyle {}_{\cos {\frac {3\pi }{8}}=\cos 67.5^{\circ }={\tfrac {\sqrt {2-{\sqrt {2}}}}{2}}}\,}$

${\displaystyle {}_{\tan {\frac {3\pi }{8}}=\tan 67.5^{\circ }={\sqrt {2}}+1}\,}$

${\displaystyle {}_{\cot {\frac {3\pi }{8}}=\cot 67.5^{\circ }={\sqrt {2}}-1}\,}$

${\displaystyle {}_{\csc {\frac {3\pi }{8}}=\csc 67.5^{\circ }={\sqrt {4-2{\sqrt {2}}}}}\,}$

${\displaystyle {}_{\sec {\frac {3\pi }{8}}=\sec 67.5^{\circ }={\sqrt {4+2{\sqrt {2}}}}}\,}$

76.5°

${\displaystyle {}_{\sin {\frac {17\pi }{40}}=\sin 76.5^{\circ }={\tfrac {{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}$

${\displaystyle {}_{\cos {\frac {17\pi }{40}}=\cos 76.5^{\circ }={\tfrac {{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}$

82.5°

${\displaystyle {}_{\sin {\frac {11\pi }{24}}=\sin 82.5^{\circ }={\tfrac {{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}+{\sqrt {12-6{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\cos {\frac {11\pi }{24}}=\cos 82.5^{\circ }={\tfrac {{\sqrt {4-2{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {12+6{\sqrt {2}}}}}{4}}}\,}$

${\displaystyle {}_{\tan {\frac {11\pi }{24}}=\tan 82.5^{\circ }={\sqrt {6}}+2+{\sqrt {5+2{\sqrt {6}}}}={\sqrt {3}}+{\sqrt {2}}+{\sqrt {6}}+2}\,}$

${\displaystyle {}_{\cot {\frac {11\pi }{24}}=\cot 82.5^{\circ }={\sqrt {6}}-2-{\sqrt {5-2{\sqrt {6}}}}={\sqrt {3}}-{\sqrt {2}}+{\sqrt {6}}-2}\,}$

cos(90/17)°

${\displaystyle {}_{\cos {\pi \over 34}=\cos {\frac {90}{17}}^{\circ }={\tfrac {\sqrt {136+6{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}}}+8{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {289+51{\sqrt {17}}-17{\sqrt {34-2{\sqrt {17}}}}-34{\sqrt {34+2{\sqrt {17}}}}}}+4{\sqrt {476+60{\sqrt {17}}-34{\sqrt {34-2{\sqrt {17}}}}-68{\sqrt {34+2{\sqrt {17}}+2{\sqrt {578-34{\sqrt {17}}}}}}-2{\sqrt {578+34{\sqrt {17}}}}}}}}{16}}}\,}$

sin(360k/17)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}65536x^{16}-278528x^{14}+487424x^{12}-452608x^{10}+239360x^{8}-71808x^{6}+11424x^{4}-816x^{2}+17=0}\,}$

${\displaystyle {}_{\sin {2\pi \over 17}=\sin {\frac {360}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {4\pi \over 17}=\sin {\frac {720}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {6\pi \over 17}=\sin {\frac {1080}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {8\pi \over 17}=\sin {\frac {1440}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {10\pi \over 17}=\sin {\frac {1800}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {12\pi \over 17}=\sin {\frac {2160}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {14\pi \over 17}=\sin {\frac {2520}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {16\pi \over 17}=\sin {\frac {2880}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {18\pi \over 17}=\sin {\frac {2880}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {20\pi \over 17}=\sin {\frac {3600}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {22\pi \over 17}=\sin {\frac {3960}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {24\pi \over 17}=\sin {\frac {4320}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {26\pi \over 17}=\sin {\frac {4680}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {28\pi \over 17}=\sin {\frac {5040}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {30\pi \over 17}=\sin {\frac {5400}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

${\displaystyle {}_{\sin {32\pi \over 17}=\sin {\frac {5760}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}$

cos(360k/17)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}256x^{8}+128x^{7}-448x^{6}-192x^{5}+240x^{4}+80x^{3}-40x^{2}-8x+1=0}\,}$

${\displaystyle {}_{\cos {2\pi \over 17}=\cos {\frac {360}{17}}^{\circ }={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {4\pi \over 17}=\cos {\frac {720}{17}}^{\circ }={\frac {-1+{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {6\pi \over 17}=\cos {\frac {1080}{17}}^{\circ }={\frac {-1-{\sqrt {17}}+{\sqrt {34+2{\sqrt {17}}}}+2{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {8\pi \over 17}=\cos {\frac {1440}{17}}^{\circ }={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {10\pi \over 17}=\cos {\frac {1800}{17}}^{\circ }={\frac {-1-{\sqrt {17}}+{\sqrt {34+2{\sqrt {17}}}}-2{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {12\pi \over 17}=\cos {\frac {2160}{17}}^{\circ }={\frac {-1-{\sqrt {17}}-{\sqrt {34+2{\sqrt {17}}}}+2{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {14\pi \over 17}=\cos {\frac {2520}{17}}^{\circ }={\frac {-1-{\sqrt {17}}-{\sqrt {34+2{\sqrt {17}}}}-2{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}$

${\displaystyle {}_{\cos {16\pi \over 17}=\cos {\frac {2880}{17}}^{\circ }={\frac {-1+{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}$

sec(360k/17)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}x^{8}-8x^{7}-40x^{6}+80x^{5}+240x^{4}-192x^{3}-448x^{2}+128x+256=0}\,}$

${\displaystyle {}_{\sec {2\pi \over 17}=\sec {\frac {360}{17}}^{\circ }={\frac {2+{\sqrt {17}}+{\sqrt {17+4{\sqrt {17}}}}-{\sqrt {34+4{\sqrt {17}}+2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {4\pi \over 17}=\sec {\frac {720}{17}}^{\circ }={\frac {2+{\sqrt {17}}-{\sqrt {17+4{\sqrt {17}}}}+{\sqrt {34+4{\sqrt {17}}-2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {6\pi \over 17}=\sec {\frac {1080}{17}}^{\circ }={\frac {2-{\sqrt {17}}+{\sqrt {17-4{\sqrt {17}}}}+{\sqrt {34-4{\sqrt {17}}+2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {8\pi \over 17}=\sec {\frac {1440}{17}}^{\circ }={\frac {2+{\sqrt {17}}+{\sqrt {17+4{\sqrt {17}}}}+{\sqrt {34+4{\sqrt {17}}+2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {10\pi \over 17}=\sec {\frac {1800}{17}}^{\circ }={\frac {2-{\sqrt {17}}+{\sqrt {17-4{\sqrt {17}}}}-{\sqrt {34-4{\sqrt {17}}+2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {12\pi \over 17}=\sec {\frac {2160}{17}}^{\circ }={\frac {2-{\sqrt {17}}-{\sqrt {17-4{\sqrt {17}}}}-{\sqrt {34-4{\sqrt {17}}-2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {14\pi \over 17}=\sec {\frac {2520}{17}}^{\circ }={\frac {2-{\sqrt {17}}-{\sqrt {17-4{\sqrt {17}}}}+{\sqrt {34-4{\sqrt {17}}-2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}$

${\displaystyle {}_{\sec {16\pi \over 17}=\sec {\frac {2880}{17}}^{\circ }={\frac {2+{\sqrt {17}}-{\sqrt {17+4{\sqrt {17}}}}-{\sqrt {34+4{\sqrt {17}}-2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}$

csc(360k/17)°

${\displaystyle {}_{\qquad {\mbox{Root of }}17x^{16}-816x^{14}+11424x^{12}-71808x^{10}+239360x^{8}-452608x^{6}+487424x^{4}-278528x^{2}+65536=0}\,}$
${\displaystyle {}_{\csc {2\pi \over 17}=\csc {\frac {360}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {4\pi \over 17}=\csc {\frac {720}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {6\pi \over 17}=\csc {\frac {1080}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {8\pi \over 17}=\csc {\frac {1440}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {10\pi \over 17}=\csc {\frac {1800}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {12\pi \over 17}=\csc {\frac {2160}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {14\pi \over 17}=\csc {\frac {2520}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {16\pi \over 17}=\csc {\frac {2880}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {18\pi \over 17}=\csc {\frac {3240}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {20\pi \over 17}=\csc {\frac {3600}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {22\pi \over 17}=\csc {\frac {3960}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {24\pi \over 17}=\csc {\frac {4320}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {26\pi \over 17}=\csc {\frac {4680}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {28\pi \over 17}=\csc {\frac {5040}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {30\pi \over 17}=\csc {\frac {5400}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

${\displaystyle {}_{\csc {32\pi \over 17}=\csc {\frac {5760}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}$

tan7.5°、tan37.5°、tan52.5°、tan82.5°

${\displaystyle {}_{\tan {\frac {\pi }{24}}=\tan 7.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}}\,}$

${\displaystyle {}_{\tan {\frac {5\pi }{24}}=\tan 37.5^{\circ }={\sqrt {6}}+{\sqrt {3}}-2-{\sqrt {2}}}\,}$

${\displaystyle {}_{\tan {\frac {7\pi }{24}}=\tan 52.5^{\circ }=2+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}\,}$

${\displaystyle {}_{\tan {\frac {11\pi }{12}}=\tan 82.5^{\circ }=2+{\sqrt {6}}+{\sqrt {2}}+{\sqrt {3}}}\,}$

sin(360k/7)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}64x^{6}-112x^{4}+56x^{2}-7=0}\,}$

${\displaystyle {}_{\sin {\frac {2\pi }{7}}=\sin {\frac {360}{7}}^{\circ }={\frac {\sqrt {7}}{6}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\sin {\frac {4\pi }{7}}=\sin {\frac {720}{7}}^{\circ }={\frac {2{\sqrt {7}}+{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{12}}}\,}$

${\displaystyle {}_{\sin {\frac {6\pi }{7}}=\sin {\frac {1080}{7}}^{\circ }=-{\frac {\sqrt {7}}{6}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\sin {\frac {8\pi }{7}}=\sin {\frac {1440}{7}}^{\circ }={\frac {\sqrt {7}}{6}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\sin {\frac {10\pi }{7}}=\sin {\frac {1800}{7}}^{\circ }=-{\frac {2{\sqrt {7}}+{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{12}}}\,}$

${\displaystyle {}_{\sin {\frac {12\pi }{7}}=\sin {\frac {2160}{7}}^{\circ }=-{\frac {\sqrt {7}}{6}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

cos(360k/7)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}8x^{3}+4x^{2}-4x-1=0}\,}$

${\displaystyle {}_{\cos {\frac {2\pi }{7}}=\cos {\frac {360}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}{12}}}\,}$

${\displaystyle {}_{\cos {\frac {4\pi }{7}}=\cos {\frac {720}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cos {\frac {6\pi }{7}}=\cos {\frac {1080}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}\,}$

tan(360k/7)

${\displaystyle {}_{\qquad {\mbox{Roots of }}x^{6}-21x^{4}+35x^{2}-7=0}\,}$

${\displaystyle {}_{\tan {\frac {2\pi }{7}}=\tan {\frac {360}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\tan {\frac {4\pi }{7}}=\tan {\frac {720}{7}}^{\circ }=-{\frac {{\sqrt {7}}+{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{3}}}\,}$

${\displaystyle {}_{\tan {\frac {6\pi }{7}}=\tan {\frac {1080}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\tan {\frac {8\pi }{7}}=\tan {\frac {1440}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\tan {\frac {10\pi }{7}}=\tan {\frac {1800}{7}}^{\circ }={\frac {{\sqrt {7}}+{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{3}}}\,}$

${\displaystyle {}_{\tan {\frac {12\pi }{7}}=\tan {\frac {2160}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}$

cot(360k/7)

${\displaystyle {}_{\qquad {\mbox{Roots of }}7x^{6}-35x^{4}+21x^{2}-1=0}\,}$

${\displaystyle {}_{\cot {\frac {2\pi }{7}}=\cot {\frac {360}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cot {\frac {4\pi }{7}}=\cot {\frac {720}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cot {\frac {6\pi }{7}}=\cot {\frac {1080}{7}}^{\circ }=-{\frac {7{\sqrt {7}}+{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}$

${\displaystyle {}_{\cot {\frac {8\pi }{7}}=\cot {\frac {1440}{7}}^{\circ }={\frac {7{\sqrt {7}}+{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}$

${\displaystyle {}_{\cot {\frac {10\pi }{7}}=\cot {\frac {1800}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\cot {\frac {12\pi }{7}}=\cot {\frac {2160}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}$

sec(360k/7)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}x^{3}+4x^{2}-4x-8=0}\,}$
${\displaystyle {}_{\sec {\frac {2\pi }{7}}=\sec {\frac {360}{7}}^{\circ }={\frac {-4+{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}{3}}}\,}$

${\displaystyle {}_{\sec {\frac {4\pi }{7}}=\sec {\frac {720}{7}}^{\circ }=-{\frac {4}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\sec {\frac {6\pi }{7}}=\sec {\frac {1080}{7}}^{\circ }=-{\frac {4}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}\,}$

csc(180/7)°

${\displaystyle {}_{\qquad {\mbox{Roots of }}7x^{6}-56x^{4}+112x^{2}-64=0}\,}$

${\displaystyle {}_{\csc {\frac {2\pi }{7}}=\csc {\frac {360}{7}}^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\csc {\frac {4\pi }{7}}=\csc {\frac {720}{7}}^{\circ }={\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\csc {\frac {6\pi }{7}}=\csc {\frac {1080}{7}}^{\circ }=-{\frac {{\sqrt[{3}]{5292{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{5292{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}$

${\displaystyle {}_{\csc {\frac {8\pi }{7}}=\csc {\frac {1440}{7}}^{\circ }={\frac {{\sqrt[{3}]{5292{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{5292{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}$

${\displaystyle {}_{\csc {\frac {10\pi }{7}}=\csc {\frac {1800}{7}}^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}$

${\displaystyle {}_{\csc {\frac {12\pi }{7}}=\csc {\frac {2160}{7}}^{\circ }={\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}$

sin(360k)°/11

${\displaystyle {}_{\qquad {\mbox{Roots of }}1024x^{10}-2816x^{8}+2816x^{6}-1232x^{4}+220x^{2}-11=0}\,}$
${\displaystyle {}_{\sin {\frac {2\pi }{11}}=\sin {\frac {360}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {4\pi }{11}}=\sin {\frac {720}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {6\pi }{11}}=\sin {\frac {1080}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {8\pi }{11}}=\sin {\frac {1440}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {10\pi }{11}}=\sin {\frac {1800}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {12\pi }{11}}=\sin {\frac {2160}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {14\pi }{11}}=\sin {\frac {2520}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {16\pi }{11}}=\sin {\frac {2880}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {18\pi }{11}}=\sin {\frac {3240}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\sin {\frac {20\pi }{11}}=\sin {\frac {3600}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

cos(360°k/11)

${\displaystyle {}_{\qquad {\mbox{Roots of }}32x^{5}+16x^{4}-32x^{3}-12x^{2}+6x+1=0}\,}$
${\displaystyle {}_{\cos {\frac {2\pi }{11}}=\cos {\frac {360}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\cos {\frac {4\pi }{11}}=\cos {\frac {720}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\cos {\frac {6\pi }{11}}=\cos {\frac {1080}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\cos {\frac {8\pi }{11}}=\cos {\frac {1440}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\cos {\frac {10\pi }{11}}=\cos {\frac {1800}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}$

${\displaystyle {}_{\qquad {\mbox{Roots of }}z^{4}+{\frac {979}{32}}z^{3}+{\frac {467181}{1024}}z^{2}+{\frac {157668929}{32768}}z+{\frac {25937424601}{1048576}}=0}\,}$
${\displaystyle {}_{z_{1}={\frac {-979+275{\sqrt {5}}+55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}$
${\displaystyle {}_{z_{2}={\frac {-979+275{\sqrt {5}}-55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}$