用户:LRT505/三角函数精确值 (非整数角度)

本表收集的是非整数角度的三角函数精确值。

注意：以下为相同角度的转换表：

相同角度的转换表

角度单位	值
转	$\mathbf {0}$	${\tfrac {1}{12}}$	${\tfrac {1}{8}}$	${\tfrac {1}{6}}$	${\tfrac {1}{4}}$	${\tfrac {1}{2}}$	${\tfrac {3}{4}}$	$\mathbf {1}$
角度	0°	30°	45°	60°	90°	180°	270°	360°
弧度	$\mathbf {0}$	${\tfrac {\pi }{6}}$	${\tfrac {\pi }{4}}$	${\tfrac {\pi }{3}}$	${\tfrac {\pi }{2}}$	$\pi$	${\tfrac {3\pi }{2}}$	$2\pi$
梯度	$0^{g}$	$33{\tfrac {1}{3}}^{g}$	$50^{g}$	$66{\tfrac {2}{3}}^{g}$	$100^{g}$	$200^{g}$	$300^{g}$	$400^{g}$

主条目：三角函数精确值

3°/16

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

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

2.5°

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

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

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

3.75°

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

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

4.5°

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

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

7.5°

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

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

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

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

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

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

11.25°

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

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

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

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

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

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

12.5°

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

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

13.5°

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

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

17.5°

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

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

22.5°

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

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

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

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

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

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

27.5°

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

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

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

31.5°

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

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

32.5°

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

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

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

33.75°

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

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

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

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

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

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

37.5°

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

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

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

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

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

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

40.5°

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

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

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

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

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

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

42.5°

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

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

49.5°

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

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

58.5°

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

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

67.5°

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

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

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

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

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

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

76.5°

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

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

82.5°

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

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

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

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

cos(90/17)°

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

sin(360k/17)°

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

cos(360k/17)°

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

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

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

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

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

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

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

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

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

sec(360k/17)°

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

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

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

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

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

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

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

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

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

csc(360k/17)°

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

${}_{\csc {2\pi \over 17}=\csc {\frac {360}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tan7.5°、tan37.5°、tan52.5°、tan82.5°

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

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

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

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

sin(360k/7)°

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

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

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

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

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

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

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

cos(360k/7)°

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

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

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

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

tan(360k/7)

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

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

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

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

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

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

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

cot(360k/7)

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

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

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

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

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

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

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

sec(360k/7)°

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

${}_{\sec {\frac {2\pi }{7}}=\sec {\frac {360}{7}}^{\circ }={\frac {-4+{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}{3}}}\,$

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

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

csc(180/7)°

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

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

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

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

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

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

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

sin(360k)°/11

${}_{\qquad {\mbox{Roots of }}1024x^{10}-2816x^{8}+2816x^{6}-1232x^{4}+220x^{2}-11=0}\,$

${}_{\sin {\frac {2\pi }{11}}=\sin {\frac {360}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

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

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

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

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

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

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

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

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

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

cos(360°k/11)

${}_{\qquad {\mbox{Roots of }}32x^{5}+16x^{4}-32x^{3}-12x^{2}+6x+1=0}\,$

${}_{\cos {\frac {2\pi }{11}}=\cos {\frac {360}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

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

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

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

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

注：该五次方程的预解方程及其根

${}_{\qquad {\mbox{Roots of }}z^{4}+{\frac {979}{32}}z^{3}+{\frac {467181}{1024}}z^{2}+{\frac {157668929}{32768}}z+{\frac {25937424601}{1048576}}=0}\,$

${}_{z_{1}={\frac {-979+275{\sqrt {5}}+55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,$

${}_{z_{2}={\frac {-979+275{\sqrt {5}}-55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,$

${}_{z_{3}={\frac {-979-275{\sqrt {5}}+55{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}{128}}}\,$

${}_{z_{4}={\frac {-979-275{\sqrt {5}}-55{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}{128}}}\,$

tan(360°k/11)

${}_{\qquad {\mbox{Roots of }}x^{10}-55x^{8}+330x^{6}-462x^{4}+165x^{2}-11=0}\,$

${}_{\tan {\frac {2\pi }{11}}=\tan {\frac {360}{11}}^{\circ }=-{\frac {3}{5}}{\sqrt {11}}+{\frac {\sqrt[{10}]{704}}{5}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {4\pi }{11}}=\tan {\frac {720}{11}}^{\circ }={\frac {3}{5}}{\sqrt {11}}-{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {6\pi }{11}}=\tan {\frac {1080}{11}}^{\circ }=-{\frac {3}{5}}{\sqrt {11}}+{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {8\pi }{11}}=\tan {\frac {1440}{11}}^{\circ }=-{\frac {3}{5}}{\sqrt {11}}+{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {10\pi }{11}}=\tan {\frac {1800}{11}}^{\circ }=-{\frac {3}{5}}{\sqrt {11}}+{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {12\pi }{11}}=\tan {\frac {2160}{11}}^{\circ }={\frac {3}{5}}{\sqrt {11}}-{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {14\pi }{11}}=\tan {\frac {2520}{11}}^{\circ }={\frac {3}{5}}{\sqrt {11}}-{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {16\pi }{11}}=\tan {\frac {2880}{11}}^{\circ }={\frac {3}{5}}{\sqrt {11}}-{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {18\pi }{11}}=\tan {\frac {3240}{11}}^{\circ }=-{\frac {3}{5}}{\sqrt {11}}+{\frac {\sqrt[{10}]{704}}{5}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\tan {\frac {20\pi }{11}}=\tan {\frac {3600}{11}}^{\circ }={\frac {3}{5}}{\sqrt {11}}-{\frac {\sqrt[{10}]{704}}{5}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

sec(360°k/11)

${}_{\qquad {\mbox{Roots of }}x^{5}+6x^{4}-12x^{3}-32x^{2}+16x+32=0}\,$

${}_{\sec {\frac {2\pi }{11}}=\sec {\frac {360}{11}}^{\circ }=-{\frac {6}{5}}+{\frac {\sqrt[{5}]{88}}{5}}\left({\frac {-3+{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3+{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1-{\sqrt {5+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1+{\sqrt {5+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {4\pi }{11}}=\sec {\frac {720}{11}}^{\circ }=-{\frac {6}{5}}+{\frac {\sqrt[{5}]{88}}{5}}\left({\frac {-3+{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3+{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1+{\sqrt {5}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1+{\sqrt {5}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {6\pi }{11}}=\sec {\frac {1080}{11}}^{\circ }=-{\frac {6}{5}}+{\frac {\sqrt[{5}]{88}}{5}}\left({\frac {1-{\sqrt {5}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1-{\sqrt {5}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3-{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3-{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {8\pi }{11}}=\sec {\frac {1440}{11}}^{\circ }=-{\frac {6}{5}}+{\frac {\sqrt[{5}]{88}}{5}}\left({\frac {1+{\sqrt {5-2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1-{\sqrt {5+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1+{\sqrt {5+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1-{\sqrt {5+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {10\pi }{11}}=\sec {\frac {1800}{11}}^{\circ }=-{\frac {6}{5}}+{\frac {\sqrt[{5}]{88}}{5}}\left({\frac {1-{\sqrt {5-2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {1+{\sqrt {5-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3-{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-3-{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,$

sin360k°/13

${}_{\qquad {\mbox{Roots of }}4096x^{12}-13312x^{10}+16640x^{8}-9984x^{6}+2912x^{4}-364x^{2}+13=0}\,$

${}_{\sin {\frac {2\pi }{13}}=\sin {\frac {360}{13}}^{\circ }={\frac {\sqrt {26-6{\sqrt {13}}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {4\pi }{13}}=\sin {\frac {720}{13}}^{\circ }={\frac {{\sqrt {26+6{\sqrt {13}}}}+{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}{12}}}\,$

${}_{\sin {\frac {6\pi }{13}}=\sin {\frac {1080}{13}}^{\circ }={\frac {{\sqrt {26-6{\sqrt {13}}}}+{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}{12}}}\,$

${}_{\sin {\frac {8\pi }{13}}=\sin {\frac {1440}{13}}^{\circ }=-{\frac {\sqrt {26-6{\sqrt {13}}}}{12}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {10\pi }{13}}=\sin {\frac {1800}{13}}^{\circ }={\frac {\sqrt {26+6{\sqrt {13}}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {12\pi }{13}}=\sin {\frac {2160}{13}}^{\circ }={\frac {\sqrt {26+6{\sqrt {13}}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {14\pi }{13}}=\sin {\frac {5040}{13}}^{\circ }=-{\frac {\sqrt {26+6{\sqrt {13}}}}{12}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {16\pi }{13}}=\sin {\frac {5400}{13}}^{\circ }=-{\frac {\sqrt {26+6{\sqrt {13}}}}{12}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {18\pi }{13}}=\sin {\frac {3240}{13}}^{\circ }={\frac {\sqrt {26-6{\sqrt {13}}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\sin {\frac {20\pi }{13}}=\sin {\frac {3600}{13}}^{\circ }=-{\frac {{\sqrt {26-6{\sqrt {13}}}}+{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}{12}}}\,$

${}_{\sin {\frac {22\pi }{13}}=\sin {\frac {3960}{13}}^{\circ }=-{\frac {{\sqrt {26+6{\sqrt {13}}}}+{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}+12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-20{\sqrt {65-18{\sqrt {13}}}}-12{\sqrt {195-54{\sqrt {13}}}}{\rm {i}}}}}{12}}}\,$

${}_{\sin {\frac {24\pi }{13}}=\sin {\frac {4320}{13}}^{\circ }=-{\frac {\sqrt {26-6{\sqrt {13}}}}{12}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}+12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-20{\sqrt {65+18{\sqrt {13}}}}-12{\sqrt {195+54{\sqrt {13}}}}{\rm {i}}}}}\,$

cos(360k°/13)

${}_{\qquad {\mbox{Roots of }}64x^{6}+32x^{5}-80x^{4}-32x^{3}+24x^{2}+6x-1=0}\,$

${}_{\cos {\frac {2\pi }{13}}=\cos {\frac {360}{13}}^{\circ }={\frac {-1+{\sqrt {13}}+{\sqrt[{3}]{104-20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\sqrt[{3}]{104-20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}{12}}}\,$

${}_{\cos {\frac {4\pi }{13}}=\cos {\frac {720}{13}}^{\circ }={\frac {-1-{\sqrt {13}}+{\sqrt[{3}]{104+20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\sqrt[{3}]{104+20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}{12}}}\,$

${}_{\cos {\frac {6\pi }{13}}=\cos {\frac {1080}{13}}^{\circ }={\frac {-1+{\sqrt {13}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104-20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104-20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}\,$

${}_{\cos {\frac {8\pi }{13}}=\cos {\frac {1440}{13}}^{\circ }={\frac {-1+{\sqrt {13}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104-20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104-20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}\,$

${}_{\cos {\frac {10\pi }{13}}=\cos {\frac {1800}{13}}^{\circ }={\frac {-1-{\sqrt {13}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104+20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104+20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}\,$

${}_{\cos {\frac {12\pi }{13}}=\cos {\frac {2160}{13}}^{\circ }={\frac {-1-{\sqrt {13}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104+20{\sqrt {13}}+12{\sqrt {39}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{104+20{\sqrt {13}}-12{\sqrt {39}}{\rm {i}}}}}\,$

csc(360k°/13)

${}_{\qquad {\mbox{Roots of }}13x^{12}-364x^{10}+2912x^{8}-9984x^{6}+16640x^{4}-13312x^{2}+4096=0}\,$

${}_{\csc {\frac {2\pi }{13}}=\csc {\frac {360}{13}}^{\circ }={\frac {13{\sqrt {26-6{\sqrt {13}}}}+{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}{39}}}\,$

${}_{\csc {\frac {4\pi }{13}}=\csc {\frac {720}{13}}^{\circ }={\frac {\sqrt {26+6{\sqrt {13}}}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {6\pi }{13}}=\csc {\frac {1080}{13}}^{\circ }={\frac {\sqrt {26-6{\sqrt {13}}}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {8\pi }{13}}=\csc {\frac {1440}{13}}^{\circ }=-{\frac {\sqrt {26-6{\sqrt {13}}}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {10\pi }{13}}=\csc {\frac {1800}{13}}^{\circ }={\frac {\sqrt {26+6{\sqrt {13}}}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {12\pi }{13}}=\csc {\frac {2160}{13}}^{\circ }={\frac {13{\sqrt {26+6{\sqrt {13}}}}+{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}{39}}}\,$

${}_{\csc {\frac {14\pi }{13}}=\csc {\frac {2520}{13}}^{\circ }=-{\frac {13{\sqrt {26+6{\sqrt {13}}}}+{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}{39}}}\,$

${}_{\csc {\frac {16\pi }{13}}=\csc {\frac {2880}{13}}^{\circ }=-{\frac {\sqrt {26+6{\sqrt {13}}}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {18\pi }{13}}=\csc {\frac {3240}{13}}^{\circ }={\frac {\sqrt {26-6{\sqrt {13}}}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {20\pi }{13}}=\csc {\frac {3600}{13}}^{\circ }=-{\frac {\sqrt {26-6{\sqrt {13}}}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {22\pi }{13}}=\csc {\frac {3960}{13}}^{\circ }=-{\frac {\sqrt {26+6{\sqrt {13}}}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}+1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{78}}{\sqrt[{3}]{338{\sqrt {12506+2106{\sqrt {13}}}}-1014{\sqrt {1014-234{\sqrt {13}}}}{\rm {i}}}}}\,$

${}_{\csc {\frac {24\pi }{13}}=\csc {\frac {4320}{13}}^{\circ }=-{\frac {13{\sqrt {26-6{\sqrt {13}}}}+{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}+1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{-338{\sqrt {12506-2106{\sqrt {13}}}}-1014{\sqrt {1014+234{\sqrt {13}}}}{\rm {i}}}}}{39}}}\,$

sec360°/13、sec720°/13、sec1080°/13、sec1440°/13、sec1800°/13、sec2160°/13

${}_{\qquad {\mbox{Roots of }}x^{6}-6x^{5}-24x^{4}+32x^{3}+80x^{2}-32x-64=0}\,$

${}_{\sec {\frac {2\pi }{13}}=\sec {\frac {360}{13}}^{\circ }={\frac {3+{\sqrt {13}}}{3}}+{\frac {3+{\sqrt {13}}}{6}}\left({\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}-442+6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}-442-6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {4\pi }{13}}=\sec {\frac {720}{13}}^{\circ }={\frac {3-{\sqrt {13}}}{3}}+{\frac {3-{\sqrt {13}}}{6}}\left({\sqrt[{3}]{126{\sqrt {13}}+442+6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{126{\sqrt {13}}+442-6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {6\pi }{13}}=\sec {\frac {1080}{13}}^{\circ }={\frac {3+{\sqrt {13}}}{3}}+{\frac {3+{\sqrt {13}}}{6}}\left({\sqrt[{3}]{126{\sqrt {13}}-442+6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}+{\sqrt[{3}]{126{\sqrt {13}}-442-6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {8\pi }{13}}=\sec {\frac {1440}{13}}^{\circ }={\frac {3+{\sqrt {13}}}{3}}+{\frac {3+{\sqrt {13}}}{6}}\left({\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}-442+6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}-442-6{\sqrt {858-234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {10\pi }{13}}=\sec {\frac {1800}{13}}^{\circ }={\frac {3-{\sqrt {13}}}{3}}+{\frac {3-{\sqrt {13}}}{6}}\left({\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}+442+6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}+442-6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

${}_{\sec {\frac {12\pi }{13}}=\sec {\frac {2160}{13}}^{\circ }={\frac {3-{\sqrt {13}}}{3}}+{\frac {3-{\sqrt {13}}}{6}}\left({\frac {1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}+442+6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{126{\sqrt {13}}+442-6{\sqrt {858+234{\sqrt {13}}}}{\rm {i}}}}\right)}\,$

cos360°/21、cos720°/21、cos1440°/21、cos1800°/21、cos2880°/21、cos3600°/21

${}_{\qquad {\mbox{Roots of }}64x^{6}-32x^{5}-96x^{4}+48x^{3}+32x^{2}-16x+1=0}\,$

${}_{\cos {\frac {2\pi }{21}}=\cos {\frac {360}{21}}^{\circ }={\frac {1+{\sqrt {21}}+{\sqrt[{3}]{154-30{\sqrt {21}}+6{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}+{\sqrt[{3}]{154-30{\sqrt {21}}-12{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}}{12}}}\,$

${}_{\cos {\frac {4\pi }{21}}=\cos {\frac {720}{21}}^{\circ }={\frac {1-{\sqrt {21}}+{\sqrt[{3}]{154+30{\sqrt {21}}+6{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}+{\sqrt[{3}]{154+30{\sqrt {21}}-12{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}}{12}}}\,$

${}_{\cos {\frac {8\pi }{21}}=\cos {\frac {1440}{21}}^{\circ }={\frac {1+{\sqrt {21}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154-30{\sqrt {21}}+6{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154-30{\sqrt {21}}-6{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\cos {\frac {10\pi }{21}}=\cos {\frac {1800}{21}}^{\circ }={\frac {1+{\sqrt {21}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154-30{\sqrt {21}}+6{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154-30{\sqrt {21}}-6{\sqrt {210-42{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\cos {\frac {16\pi }{21}}=\cos {\frac {2880}{21}}^{\circ }={\frac {1-{\sqrt {21}}}{12}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154+30{\sqrt {21}}+6{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154+30{\sqrt {21}}-6{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\cos {\frac {20\pi }{21}}=\cos {\frac {3600}{21}}^{\circ }={\frac {1-{\sqrt {21}}}{12}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154+30{\sqrt {21}}+6{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{154+30{\sqrt {21}}-6{\sqrt {210+42{\sqrt {21}}}}{\rm {i}}}}}\,$

sec120°/7、cos240°/7、sec480°/7、sec600°/7、sec960°/7、sec1200°/7

${}_{\qquad {\mbox{Roots of }}x^{6}-16x^{5}+32x^{4}+48x^{3}-96x^{2}-32x+64=0}\,$

${}_{\sec {\frac {2\pi }{21}}=\sec {\frac {120}{7}}^{\circ }={\frac {8+2{\sqrt {21}}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728+156{\sqrt {21}}+12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728+156{\sqrt {21}}-12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\sec {\frac {4\pi }{21}}=\sec {\frac {240}{7}}^{\circ }={\frac {8-2{\sqrt {21}}+{\sqrt[{3}]{728-156{\sqrt {21}}+12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}+{\sqrt[{3}]{728-156{\sqrt {21}}-12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}}{3}}}\,$

${}_{\sec {\frac {8\pi }{21}}=\sec {\frac {480}{7}}^{\circ }={\frac {8+2{\sqrt {21}}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728+156{\sqrt {21}}+12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728+156{\sqrt {21}}-12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\sec {\frac {10\pi }{21}}=\sec {\frac {600}{7}}^{\circ }={\frac {8+2{\sqrt {21}}+{\sqrt[{3}]{728+156{\sqrt {21}}+12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}+{\sqrt[{3}]{728+156{\sqrt {21}}-12{\sqrt {1155+252{\sqrt {21}}}}{\rm {i}}}}}{3}}}\,$

${}_{\sec {\frac {16\pi }{21}}=\sec {\frac {960}{7}}^{\circ }={\frac {8-2{\sqrt {21}}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728-156{\sqrt {21}}+12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728-156{\sqrt {21}}-12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}}\,$

${}_{\sec {\frac {20\pi }{21}}=\sec {\frac {1200}{7}}^{\circ }={\frac {8-2{\sqrt {21}}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728-156{\sqrt {21}}+12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{728-156{\sqrt {21}}-12{\sqrt {1155-252{\sqrt {21}}}}{\rm {i}}}}}\,$

cos(2kpi/19)

${}_{\qquad {\mbox{Roots of }}512x^{9}+256x^{8}-1024x^{7}-448x^{6}+672x^{5}+240x^{4}-160x^{3}-40x^{2}+10x+1=0}\,$

${}_{\cos {\frac {2\pi }{19}}=\cos {\frac {360}{19}}^{\circ }=-{\frac {1}{18}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}+{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}-{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {1}{2}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}+{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}+{\frac {1}{2}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}-{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}}\,$

${}_{\cos {\frac {14\pi }{19}}=\cos {\frac {2520}{19}}^{\circ }=-{\frac {1}{18}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}+{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}-{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{4}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}+{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{4}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}-{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}}\,$

${}_{\cos {\frac {16\pi }{19}}=\cos {\frac {2880}{19}}^{\circ }=-{\frac {1}{18}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}+{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{{\frac {133}{54}}-{\frac {19}{18}}{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{4}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}+{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{4}}{\sqrt[{3}]{-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}-{\sqrt {19\left[-{\frac {\sqrt[{3}]{361}}{27}}+{\frac {2}{27}}(-1-{\sqrt {3}}{\rm {i}}){\frac {\sqrt[{3}]{19}}{\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}}+{\frac {5}{108}}(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{-28+12{\sqrt {3}}{\rm {i}}}}\right]^{3}-\left[-{\frac {38}{81}}+(-1-{\sqrt {3}}{\rm {i}}){\frac {7{\sqrt[{3}]{361}}}{3{\sqrt[{3}]{44+282{\sqrt {3}}{\rm {i}}}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{12}}{\sqrt[{3}]{836+7068{\sqrt {3}}{\rm {i}}}}-(-1-{\sqrt {3}}{\rm {i}}){\frac {31{\sqrt[{3}]{361}}}{81{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}}}-{\frac {-1+{\sqrt {3}}{\rm {i}}}{324}}{\sqrt[{3}]{51148+59052{\sqrt {3}}{\rm {i}}}}\right]^{2}}}{\rm {i}}}}}\,$

cos(2pi/23)

${}_{\qquad {\mbox{Root of }}2048x^{11}+1024x^{10}-5120x^{9}-2304x^{8}+4608x^{7}+1792x^{6}-1792x^{5}-560x^{4}+280x^{3}+60x^{2}-12x-1=0}\,$

${}_{\cos {\frac {2\pi }{23}}=\cos {\frac {360}{23}}^{\circ }=-{\frac {1}{22}}+{\sqrt[{11}]{2783}}\cdot {\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {25178-2596{\sqrt {5}}+\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25178-2596{\sqrt {5}}-\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}+\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}-\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {25178-2596{\sqrt {5}}+\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25178-2596{\sqrt {5}}-\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}+\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}-\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot {\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {25178-2596{\sqrt {5}}+\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25178-2596{\sqrt {5}}-\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}+\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}-\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}-{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {25178-2596{\sqrt {5}}+\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25178-2596{\sqrt {5}}-\left(12567{\sqrt {10+2{\sqrt {5}}}}+783{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}+\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878-12444{\sqrt {5}}-\left(7973{\sqrt {10+2{\sqrt {5}}}}+6037{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)-{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-546-9302{\sqrt {5}}+\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546-9302{\sqrt {5}}-\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}+\left(5377{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}-\left(5277{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-546-9302{\sqrt {5}}+\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546-9302{\sqrt {5}}-\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}+\left(5377{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}-\left(5277{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)+{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-546-9302{\sqrt {5}}+\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546-9302{\sqrt {5}}-\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}+\left(5377{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}-\left(5277{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}-{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-546-9302{\sqrt {5}}+\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546-9302{\sqrt {5}}-\left(-3595{\sqrt {10+2{\sqrt {5}}}}+330{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}+\left(5377{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957-1443{\sqrt {5}}-\left(5277{\sqrt {10+2{\sqrt {5}}}}+5254{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)-{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-40957+1443{\sqrt {5}}+\left(5254{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957+1443{\sqrt {5}}-\left(5254{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}-\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}+\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-40957+1443{\sqrt {5}}+\left(5257{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957+1443{\sqrt {5}}-\left(5257{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}-\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25183+2596{\sqrt {5}}+\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)+{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-40957+1443{\sqrt {5}}+\left(5254{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957+1443{\sqrt {5}}-\left(5254{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}-\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}+\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}-{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-40957+1443{\sqrt {5}}+\left(5257{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-40957+1443{\sqrt {5}}-\left(5257{\sqrt {10+2{\sqrt {5}}}}+5377{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25187+2596{\sqrt {5}}-\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {25183+2596{\sqrt {5}}+\left(783{\sqrt {10+2{\sqrt {5}}}}+12567{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)-{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {21878+12444{\sqrt {5}}-\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878+12444{\sqrt {5}}+\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}-\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}+\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {21878+12444{\sqrt {5}}-\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878+12444{\sqrt {5}}+\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}-\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}+\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)+{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {21878+12444{\sqrt {5}}-\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878+12444{\sqrt {5}}+\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}-\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}+\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}-{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {21878+12444{\sqrt {5}}-\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {21878+12444{\sqrt {5}}+\left(6037{\sqrt {10+2{\sqrt {5}}}}-7973{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}-\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007-7313{\sqrt {5}}+\left(13227{\sqrt {10+2{\sqrt {5}}}}-4594{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)+{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-5007+7313{\sqrt {5}}-\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007+7313{\sqrt {5}}+\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}+\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}-\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-5007+7313{\sqrt {5}}-\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007+7313{\sqrt {5}}+\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}+\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}-\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

${}_{+{\sqrt[{11}]{2783}}\cdot \left[-{\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)-{\rm {i}}{\sqrt {{\frac {11}{20}}+{\frac {\sqrt[{5}]{88}}{40}}\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)}}\right]{\sqrt[{11}]{-{\frac {2300783}{1210}}+{\frac {-5007+7313{\sqrt {5}}-\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007+7313{\sqrt {5}}+\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}+\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}-\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}-{\sqrt {{\frac {1801152661463}{14641}}-\left(-{\frac {2300783}{1210}}+{\frac {-5007+7313{\sqrt {5}}-\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-5007+7313{\sqrt {5}}+\left(4594{\sqrt {10+2{\sqrt {5}}}}+13227{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}+\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-546+9302{\sqrt {5}}-\left(330{\sqrt {10+2{\sqrt {5}}}}+3595{\sqrt {10-2{\sqrt {5}}}}\right){\rm {i}}}{2}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}\right)^{2}}}{\rm {i}}}}}\,$

sec(2kπ/27)

${}_{\qquad {\mbox{Roots of }}x^{9}+18x^{8}-240x^{6}+864x^{4}-1152x^{2}+512=0}\,$

${}_{\sec {\frac {2\pi }{27}}=\sec {\frac {40}{3}}^{\circ }=-2+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4+4{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4-4{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}+2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}+{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}-2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}}\,$

${}_{\sec {\frac {16\pi }{27}}=\sec {\frac {320}{3}}^{\circ }=-2+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4+4{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4-4{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}+2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}-2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}}\,$

${}_{\sec {\frac {20\pi }{27}}=\sec {\frac {400}{3}}^{\circ }=-2+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4+4{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-4-4{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}+2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{-60+2{\sqrt[{3}]{-26636+292{\sqrt {3}}{\rm {i}}}}+2{\sqrt[{3}]{-26636-292{\sqrt {3}}{\rm {i}}}}-2{\sqrt {356+(-1-{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724+8424{\sqrt {3}}{\rm {i}}}}+(-1+{\sqrt {3}}{\rm {i}}){\sqrt[{3}]{5607724-8424{\sqrt {3}}{\rm {i}}}}}}{\rm {i}}}}}\,$

sin(kπ/120)

${}_{\qquad {\mbox{Roots of }}4294967296x^{32}-34359738368x^{30}+124554051584x^{28}-270582939648x^{26}+392603631616x^{24}-401411670016x^{22}+297388736512x^{20}-161694613504x^{18}+64647462912x^{16}-18868600832x^{14}+3953983488x^{12}-578486272x^{10}+56539136x^{8}-3436544x^{6}+114176x^{4}-1536x^{2}+1=0}\,$

${}_{\sin {\frac {\pi }{120}}=\sin 1.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,$

${}_{\sin {\frac {7\pi }{120}}=\sin 10.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}-{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}+4{\sqrt {5}}+2{\sqrt {10}}}}+{\sqrt {60-6{\sqrt {10}}-30{\sqrt {2}}+12{\sqrt {5}}}}+{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,$

${}_{\sin {\frac {11\pi }{120}}=\sin 16.5^{\circ }={\frac {{\sqrt {30-15{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {10-2{\sqrt {5}}+5{\sqrt {2}}-{\sqrt {10}}}}-2{\sqrt {30-6{\sqrt {5}}-15{\sqrt {2}}+3{\sqrt {10}}}}}{16}}}\,$

${}_{\sin {\frac {59\pi }{120}}=\sin 88.5^{\circ }={\frac {{\sqrt {30-15{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {20-10{\sqrt {2}}-4{\sqrt {5}}+2{\sqrt {10}}}}+{\sqrt {60-6{\sqrt {10}}+30{\sqrt {2}}-12{\sqrt {5}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{16}}}\,$

${}_{\sin {\frac {119\pi }{120}}=\sin 178.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,$

tan[k(arctan4)]

${}_{\tan {\frac {\arctan 4}{4}}={\frac {{\sqrt {34+2{\sqrt {17}}}}-{\sqrt {17}}-1}{4}}}\,$

${}_{\tan {\frac {\arctan 4}{2}}={\frac {{\sqrt {17}}-1}{4}}}\,$

${}_{\tan {\frac {3}{4}}\arctan 4={\frac {17{\sqrt {17}}-47+{\sqrt {9826-1598{\sqrt {17}}}}}{52}}}\,$

${}_{\tan {\frac {5}{4}}\arctan 4=-{\frac {1121+289{\sqrt {17}}+{\sqrt {2839714+647938{\sqrt {17}}}}}{404}}}\,$

${}_{\tan {\frac {3}{2}}\arctan 4=-{\frac {47+17{\sqrt {17}}}{52}}}\,$

${}_{\tan {\frac {7}{4}}\arctan 4=-{\frac {20047+4913{\sqrt {17}}+17{\sqrt {2839714+681598{\sqrt {17}}}}}{2908}}}\,$

${}_{\tan {\frac {\arctan 4}{3}}=4+{\frac {-1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{68+17{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{68-17{\rm {i}}}}}\,$

${}_{\tan {\frac {2}{3}}\arctan 4={\frac {-8+{\sqrt[{3}]{-2312+4335{\rm {i}}}}+{\sqrt[{3}]{-2312-4335{\rm {i}}}}}{15}}}\,$