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：
| 角度单位 | 值 | |||||||
|---|---|---|---|---|---|---|---|---|
| 转 | ||||||||
| 角度 | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| 弧度 | ||||||||
| 梯度 | ||||||||
3°/16
编辑
2.5°
编辑![{\displaystyle {}_{\sin {\frac {\pi }{72}}=\sin 2.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f434868a04a142384fdc73caec57014d50c0d75)
![{\displaystyle {}_{\cos {\frac {\pi }{72}}=\cos 2.5^{\circ }={\tfrac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/866f67f49d41fa24b436cd1a4e93414c6e2574e3)
![{\displaystyle {}_{\tan {\frac {\pi }{72}}=\tan 2.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}-{\frac {1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}-{\frac {1-{\sqrt {3}}{\mathrm {i} }}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66128b827d1ec80892266276423604452cc9bb35)
3.75°
编辑
4.5°
编辑
7.5°
编辑
11.25°
编辑
12.5°
编辑![{\displaystyle {}_{\sin {\frac {5\pi }{72}}=\sin 12.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44422106c1a592643558231da2417f950cabfc63)
![{\displaystyle {}_{\cos {\frac {5\pi }{72}}=\cos 12.5^{\circ }={\tfrac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\mathrm {i} }}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\mathrm {i} }}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76abc2f308a97308e48a1a52066c33856d8465ee)
13.5°
编辑
17.5°
编辑![{\displaystyle {}_{\sin {\frac {7\pi }{72}}=\sin 17.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2419afa05c267ca670f6f145e95fab66e59ec558)
![{\displaystyle {}_{\cos {\frac {7\pi }{72}}=\cos 17.5^{\circ }={\frac {{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/086d3dec7156ad5fcf46ca505c0adb155074e669)
22.5°
编辑
27.5°
编辑![{\displaystyle {}_{\sin {\frac {11\pi }{72}}=\sin 27.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d417bfaf943d0e904edebe69db148237403759c)
![{\displaystyle {}_{\cos {\frac {11\pi }{72}}=\cos 27.5^{\circ }={\frac {{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59cb3c195ca7dd7abab5d3a76228ad781823519f)
![{\displaystyle {}_{\cot {\frac {11\pi }{72}}=\cot 27.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}+{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}+{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc66c230fdbbe019660f65b84edad06f15a5fd9)
31.5°
编辑
32.5°
编辑![{\displaystyle {}_{\sin {\frac {13\pi }{72}}=\sin 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359e32692beaf9fa857dd7b1710b46b041b126f3)
![{\displaystyle {}_{\cos {\frac {13\pi }{72}}=\cos 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819a5c13523fa06c7732d30654f254f67d868f63)
![{\displaystyle {}_{\cot {\frac {13\pi }{72}}=\cot 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112+{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{2}}{\sqrt[{3}]{46{\sqrt {6}}+78{\sqrt {2}}-64{\sqrt {3}}-112-{\sqrt {30318{\sqrt {6}}+35186{\sqrt {2}}-49766}}{\rm {i}}}}-{\sqrt {6}}-{\sqrt {2}}+2+{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf971ccb6e38653149b98fce3e847ff360a829d)
33.75°
编辑
37.5°
编辑
40.5°
编辑
42.5°
编辑![{\displaystyle {}_{\sin {\frac {17\pi }{72}}=\sin 42.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c88211f1cc708883b46a11e659cdd449cdcc689)
![{\displaystyle {}_{\cos {\frac {17\pi }{72}}=\cos 42.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a31a07b223382c38eaa490ddcf7ea591fde19a1)
49.5°
编辑
58.5°
编辑
67.5°
编辑
76.5°
编辑
82.5°
编辑
cos(90/17)°
编辑
sin(360k/17)°
编辑
cos(360k/17)°
编辑
sec(360k/17)°
编辑
csc(360k/17)°
编辑
tan7.5°、tan37.5°、tan52.5°、tan82.5°
编辑
sin(360k/7)°
编辑
![{\displaystyle {}_{\sin {\frac {2\pi }{7}}=\sin {\frac {360}{7}}^{\circ }={\frac {\sqrt {7}}{6}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/443d7c34965a1659a3b517c3e4a3483da4dfd2b6)
![{\displaystyle {}_{\sin {\frac {4\pi }{7}}=\sin {\frac {720}{7}}^{\circ }={\frac {2{\sqrt {7}}+{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{12}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d16f57681a636c2983367966112cffde8087a823)
![{\displaystyle {}_{\sin {\frac {6\pi }{7}}=\sin {\frac {1080}{7}}^{\circ }=-{\frac {\sqrt {7}}{6}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118cfb40686131b3d6c0d915a20fcd7c90f683d0)
![{\displaystyle {}_{\sin {\frac {8\pi }{7}}=\sin {\frac {1440}{7}}^{\circ }={\frac {\sqrt {7}}{6}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed53db98db2b40f1f028b35b38b114f53f726728)
![{\displaystyle {}_{\sin {\frac {10\pi }{7}}=\sin {\frac {1800}{7}}^{\circ }=-{\frac {2{\sqrt {7}}+{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{12}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77b7cf595e8ee776b2cc175eb3608f60d542720c)
![{\displaystyle {}_{\sin {\frac {12\pi }{7}}=\sin {\frac {2160}{7}}^{\circ }=-{\frac {\sqrt {7}}{6}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{-52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f001241ce7179864c734bf5c1954bb6a3d12aa)
cos(360k/7)°
编辑
![{\displaystyle {}_{\cos {\frac {2\pi }{7}}=\cos {\frac {360}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}{12}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda95661a453df96950fd00fb52aa3f107819c95)
![{\displaystyle {}_{\cos {\frac {4\pi }{7}}=\cos {\frac {720}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118186ca0a96c38426e34d4529dc554efd1c5b28)
![{\displaystyle {}_{\cos {\frac {6\pi }{7}}=\cos {\frac {1080}{7}}^{\circ }=-{\frac {1}{6}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{24}}{\sqrt[{3}]{28-84{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bfc4e952997e2fd3016d854f41a7a9a687774ef)
tan(360k/7)
编辑
![{\displaystyle {}_{\tan {\frac {2\pi }{7}}=\tan {\frac {360}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe5c75a54a1db7eb7a06062de83ea10389ede32)
![{\displaystyle {}_{\tan {\frac {4\pi }{7}}=\tan {\frac {720}{7}}^{\circ }=-{\frac {{\sqrt {7}}+{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2de84d5b56f4df65febf4594459538c7b9d9b4ed)
![{\displaystyle {}_{\tan {\frac {6\pi }{7}}=\tan {\frac {1080}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65259264cf216fdd7b3422ef63a27653bb01d1b4)
![{\displaystyle {}_{\tan {\frac {8\pi }{7}}=\tan {\frac {1440}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/298a1497d5ac0899eab4b02072af600d11705417)
![{\displaystyle {}_{\tan {\frac {10\pi }{7}}=\tan {\frac {1800}{7}}^{\circ }={\frac {{\sqrt {7}}+{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}{3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a75fc172e4e20a1cf972bcae6c2cb3535fce95)
![{\displaystyle {}_{\tan {\frac {12\pi }{7}}=\tan {\frac {2160}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}+12{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{52{\sqrt {7}}-12{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9356527801576d447307b96ea7b768ed3da66df2)
cot(360k/7)
编辑
![{\displaystyle {}_{\cot {\frac {2\pi }{7}}=\cot {\frac {360}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c935ddeaf9932facc86fe5ac41cb09b850ce79)
![{\displaystyle {}_{\cot {\frac {4\pi }{7}}=\cot {\frac {720}{7}}^{\circ }={\frac {\sqrt {7}}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e72a0c3f4443b1bd38bef4aa1ccd84c2860b01)
![{\displaystyle {}_{\cot {\frac {6\pi }{7}}=\cot {\frac {1080}{7}}^{\circ }=-{\frac {7{\sqrt {7}}+{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f21670bf5faff2d23b74e5e7540dc9ecbfdad982)
![{\displaystyle {}_{\cot {\frac {8\pi }{7}}=\cot {\frac {1440}{7}}^{\circ }={\frac {7{\sqrt {7}}+{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5a6b69b0f0e10d31089df221c65bd54d7166cc0)
![{\displaystyle {}_{\cot {\frac {10\pi }{7}}=\cot {\frac {1800}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d22bbbc03ed61b27cdd251d047f2900ecf8f1103)
![{\displaystyle {}_{\cot {\frac {12\pi }{7}}=\cot {\frac {2160}{7}}^{\circ }=-{\frac {\sqrt {7}}{3}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{196{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2fa89a3f40b7da08c71f618a7b6768ca0446687)
sec(360k/7)°
编辑
![{\displaystyle {}_{\sec {\frac {2\pi }{7}}=\sec {\frac {360}{7}}^{\circ }={\frac {-4+{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}{3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe71b7a99f3d65e86f7eacb180d6102a574ee28c)
![{\displaystyle {}_{\sec {\frac {4\pi }{7}}=\sec {\frac {720}{7}}^{\circ }=-{\frac {4}{3}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba48015ec4a7313c76dd053d69e8b79c076ed4f)
![{\displaystyle {}_{\sec {\frac {6\pi }{7}}=\sec {\frac {1080}{7}}^{\circ }=-{\frac {4}{3}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28+84{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{6}}{\sqrt[{3}]{-28-84{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b346d633cddc35a3f7dc3f17f824c3274029f1)
csc(180/7)°
编辑
![{\displaystyle {}_{\csc {\frac {2\pi }{7}}=\csc {\frac {360}{7}}^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8584951d904882993268c9f4f2d7eb5614b71187)
![{\displaystyle {}_{\csc {\frac {4\pi }{7}}=\csc {\frac {720}{7}}^{\circ }={\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99bd969d1fe857948663647c673e1b40f2e03e33)
![{\displaystyle {}_{\csc {\frac {6\pi }{7}}=\csc {\frac {1080}{7}}^{\circ }=-{\frac {{\sqrt[{3}]{5292{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{5292{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/005bf8ceaf1d7565d665f6ccd24eb3e74ab3ae82)
![{\displaystyle {}_{\csc {\frac {8\pi }{7}}=\csc {\frac {1440}{7}}^{\circ }={\frac {{\sqrt[{3}]{5292{\sqrt {7}}+588{\sqrt {21}}{\rm {i}}}}+{\sqrt[{3}]{5292{\sqrt {7}}-588{\sqrt {21}}{\rm {i}}}}}{21}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bcb0ee64223a91946d4bb9e41776213b83216b9)
![{\displaystyle {}_{\csc {\frac {10\pi }{7}}=\csc {\frac {1800}{7}}^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15f8df85aa456ad875aa39890857cd4ca694af69)
![{\displaystyle {}_{\csc {\frac {12\pi }{7}}=\csc {\frac {2160}{7}}^{\circ }={\frac {-1+{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292+588{\sqrt {3}}{\rm {i}}}}+{\frac {-1-{\sqrt {3}}{\rm {i}}}{42}}{\sqrt[{3}]{5292-588{\sqrt {3}}{\rm {i}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd9fab0a0104c24bef07345c523960e62261f904)
sin(360k)°/11
编辑
![{\displaystyle {}_{\sin {\frac {2\pi }{11}}=\sin {\frac {360}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c779eaf62a34c9fb3894279066cdf563dddf2b)
![{\displaystyle {}_{\sin {\frac {4\pi }{11}}=\sin {\frac {720}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3309bf7e8f62cae62764ea7f795d389d82ed03a)
![{\displaystyle {}_{\sin {\frac {6\pi }{11}}=\sin {\frac {1080}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f05b2bccedc7a5819afcd76db5d8bbe347237fe)
![{\displaystyle {}_{\sin {\frac {8\pi }{11}}=\sin {\frac {1440}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79077d975656ff371e4360d1b67f8f9720b8023a)
![{\displaystyle {}_{\sin {\frac {10\pi }{11}}=\sin {\frac {1800}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b59badd50867f21e314987750f4ce8523992aaad)
![{\displaystyle {}_{\sin {\frac {12\pi }{11}}=\sin {\frac {2160}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3846f0c270e4d55d00c3eb89a94f1f81bb267084)
![{\displaystyle {}_{\sin {\frac {14\pi }{11}}=\sin {\frac {2520}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5178d0a3978adeb01aa5dfe3c214bbe45384080f)
![{\displaystyle {}_{\sin {\frac {16\pi }{11}}=\sin {\frac {2880}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1115eb6a3185de326a62200016da83a4bdcaa20)
![{\displaystyle {}_{\sin {\frac {18\pi }{11}}=\sin {\frac {3240}{11}}^{\circ }={\frac {\sqrt {11}}{10}}+{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c4af9cefe8831526e041b7cc272770c2747ddf)
![{\displaystyle {}_{\sin {\frac {20\pi }{11}}=\sin {\frac {3600}{11}}^{\circ }=-{\frac {\sqrt {11}}{10}}-{\frac {\sqrt[{10}]{704}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}+5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{109+25{\sqrt {5}}-5{\sqrt {8770-218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}+5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{109-25{\sqrt {5}}-5{\sqrt {8770+218{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca781397412e7c457488c0ddb6f3ae99f29b31c)
cos(360°k/11)
编辑
![{\displaystyle {}_{\cos {\frac {2\pi }{11}}=\cos {\frac {360}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e085e77bde6ad9c25220634a0105ad662edfbdd5)
![{\displaystyle {}_{\cos {\frac {4\pi }{11}}=\cos {\frac {720}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9282175548232077e4085e14876ecdd9feb772d)
![{\displaystyle {}_{\cos {\frac {6\pi }{11}}=\cos {\frac {1080}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75109c34023bece29efa53495f34b3fb87fc4b22)
![{\displaystyle {}_{\cos {\frac {8\pi }{11}}=\cos {\frac {1440}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}+{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1+{\sqrt {5}}-{\sqrt {10+2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68d0d870c076060b0aad5e8a876c856232cf6f9d)
![{\displaystyle {}_{\cos {\frac {10\pi }{11}}=\cos {\frac {1800}{11}}^{\circ }=-{\frac {1}{10}}+{\frac {\sqrt[{5}]{88}}{20}}\left({\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89+5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{25{\sqrt {5}}-89-5{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}-{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89+5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}+{\frac {-1-{\sqrt {5}}+{\sqrt {10-2{\sqrt {5}}}}{\rm {i}}}{4}}{\sqrt[{5}]{-25{\sqrt {5}}-89-5{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/690bb54c55da729ffc6847bcce0c11ba20014959)
注：该五次方程的预解方程及其根
tan(360°k/11)
编辑
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3cf95987ec86c08c3222cd4dc7e071781134ce)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa4be47025295ed34d0a0b9f8f2b0e2aef010da)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1089d4fb5d44840fc2c4b52cea6dfa77bc468ba)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/397c655317e25aa3199109c48d186822c706303a)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a82b7b1ff5cacdd85c1e166c6c1c7a6960f7d68f)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a507c9e0c8addf768852fe30c6f44b7b76c52d1a)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e41cbb6787a304d1c4dc6248e4d72eb553ca92e)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/803c266231789664a8a3fc7c24b65548c6d5fced)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/402598ac31b0a416a64f53b23d0585e57bc574f8)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68325ee0db69f28aa985380bc13ebecf411cfebb)
sec(360°k/11)
编辑
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47590101978810dadf9ba46964f215121a868c47)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b51df23cecb912996a1450fc41f0271980f1906)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1be31fbc64b9803d6b0295198b7e18c94883b9)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87af5f0d3c9229edd1ede5f687f85196aa5f6b8c)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53e6155322c688891fe58e43b48fc2a00831d656)
sin360k°/13
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57ef36bdded89392be4a60f38103586fc8c17098)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a57cff9daee8a5a9ff11056b84d221ac0f6b1e93)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed6160e14a0d791a88a3e1b0d9ec800fd69a1b2)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa23fd681348853827cb4e105337bb14bd2c7db1)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66ab34f5f51f98a112c57c493781e62c542bac69)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d1b18c40456dfcf7f8ea2c766b0b5fbaf37a7ddc)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1422652f57562ad8d1fb3dfb6ad457088bd51df)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5936a4c11153460223d67d4c8385ebf464d68e)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/def69eaf057b28d7daf78c4213dc9deed11a8ac5)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a519db64ae41f6db9e6856044bf7ca2bed2f869)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6405850cb6545a2b3f0c8a71376fa2bf41cedc30)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26964dcd770173f503a7c5b29612c19c4cfa425f)
cos(360k°/13)
编辑
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6250740f2bb483b42612d318fb6e252d4d8bf8d)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92f5e8a1e7a751bbd606b60b088ee1e9f0e2ed26)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b229d0ffd1ba6ff06e749eb77d62809cb17603b8)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b423568118591b8c625b202d166267d56fed6cff)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/130449989a7211c0349837a1610eb276510deb17)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e25a4fe09d88fbeb57966fdfda10d737bd01bf8a)
csc(360k°/13)
编辑
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21cfafae57d2ff80a29cfa810f2151191fbf884d)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddfb2178998307532e500dfbe8e3652ca3caf867)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa038e3eb650fe6c4da20decf3a3cd56a594d47e)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2515b3b1534894b0222257dc19cd0be6db45c74)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e2f8b51a9d54dc3d2606ed09e5b7faa4c92a93)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e574367a2744a56aaef3787bb57088429eefa2a6)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e422176c42a4bc80ecfdda25cde5332775aa377c)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7a8a2391dc09a4bdd3dc894f1c35fddc22fb4a)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51864156a8bac2c33cbbc16bb48ee19c5ff0b6a9)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f494548937a0981f6b78b60445b60a58680a63b)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3ecd615eb77ed11c6a4db5b8459744403ad56c)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba286f8a7ac6705e15001c883816874221d3f96b)
sec360°/13、sec720°/13、sec1080°/13、sec1440°/13、sec1800°/13、sec2160°/13
编辑
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04f2ee5e4f648e641212327b29dac597f1cdbb9e)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a986709fb011f4d5803db83675cb4b4f91b36d11)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42676ce3f3358e43da43b5464ffa67bc2968b06c)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8236bc07cf939cf4132aef0625907f9950febe)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66c01f4f05958c2c43cd4b3461389e2edffec811)
![{\displaystyle {}_{\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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb191f98b7beb635fa1d88f6db3f76b0d45aabd5)
cos360°/21、cos720°/21、cos1440°/21、cos1800°/21、cos2880°/21、cos3600°/21
编辑
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10db470b9719e0dc243ce560c78a7a1f3ff9200)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e51f105efc4e343114eed5f39223415c19e0e8b)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36cdd27dc4822d91ee0175f8f22184a5fc53e74d)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2931e7f6230787aeb9d63e523fa66654408dfa)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41fdcf4d438cbdb02c2e7fb598389a0bfd802a3d)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63b545f15a18143ceb4c415f4db28b49a12aac00)
sec120°/7、cos240°/7、sec480°/7、sec600°/7、sec960°/7、sec1200°/7
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0960924a7a038c5cab28dde7dfc902e510c21504)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b76006162a8911911f024e9c5b29be4544a23a)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecfcded8515c937e042be7c96441f6d3b4ff8cc)
![{\displaystyle {}_{\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}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66764f19898d9807836d9e518cfe3b989e4bc6dd)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6495505666c01bef42db7d726af3a44678779f1d)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc498775cf94803db9c12920ec6c0edf62d2eb2b)
cos(2kpi/19)
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/691fe64f59009fb29f576115c7288feedb12aba8)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e9a2296878b86892030fe73717d7b7271ba74c)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79a5532b21078e18e8219072ec25563e22e085b0)
cos(2pi/23)
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fee35240b2bdda5f8594dcf83576e19627747efa)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05adb8309ac7717ba0e8f660aa96212dc2da4725)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/889ac9f098e12254ee6b52f03142c15835307509)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0069e817fa6428307cc5b0a0240dd010ac1c0f7a)
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![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98ed6c2c0b9c6bea2df4f9994897e6fdb2413fc5)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1fa5f74e1b51b1d8557f30135685c40f4ca200f)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed53c55e41c5ccd8d27a51d8cf1af0f6951dcd30)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b90dddc116891c9b9611f201ce378c1231169e9e)
![{\displaystyle {}_{+{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ac0b7bbfbcb5dbd84360b61f59898f71a2c491)
sec(2kπ/27)
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b022dd647b0924ae70a5e3fc92c1b82d164cc4b8)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bebccb3e8c3601260571dad7761232f1e7495176)
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25def9c48d1638e9990924aff7008033296fac95)
sin(kπ/120)
编辑
tan[k(arctan4)]
编辑
![{\displaystyle {}_{\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}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a781f8af465829fde3bb6462c40508985603c57)
![{\displaystyle {}_{\tan {\frac {2}{3}}\arctan 4={\frac {-8+{\sqrt[{3}]{-2312+4335{\rm {i}}}}+{\sqrt[{3}]{-2312-4335{\rm {i}}}}}{15}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/695c96a69b3788dc182ea696be8129ece8ff4188)