本表收集的是非整數角度的三角函數精確值。
注意:以下為相同角度的轉換表:
相同角度的轉換表
角度單位 |
值
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轉
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角度
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0°
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30°
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45°
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60°
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90°
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180°
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270°
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360°
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弧度
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梯度
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3°/16
![{\displaystyle {}_{\sin {\frac {\pi }{960}}=\sin 0.1875^{\circ }={\frac {\sqrt {8-{\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+2{\sqrt {3}}+2{\sqrt {15}}+2{\sqrt {10-2{\sqrt {5}}}}}}}}}}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e10b1aa2a52bae6704da2c0cb07c0be385279b)
![{\displaystyle {}_{\cos {\frac {\pi }{960}}=\cos 0.1875^{\circ }={\frac {\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+{\sqrt {8+2{\sqrt {3}}+2{\sqrt {15}}+2{\sqrt {10-2{\sqrt {5}}}}}}}}}}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa4e24e4a36cbcb406f8fb73e89def306ea08b3)
2.5°
![{\displaystyle {}_{\sin {\frac {\pi }{72}}=\sin 2.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {\pi }{48}}=\sin 3.75^{\circ }={\tfrac {{\sqrt {6+3{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12+6{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12+6{\sqrt {2+{\sqrt {2}}}}+6{\sqrt {2}}+3{\sqrt {4+2{\sqrt {2}}}}}}-{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {2}}-{\sqrt {4+2{\sqrt {2}}}}}}}{4}}={\tfrac {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9aa0188fd1ce57f8fa6ca2b462d7e726fb0810)
![{\displaystyle {}_{\cos {\frac {\pi }{48}}=\cos 3.75^{\circ }={\tfrac {{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}}}-{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {2}}+{\sqrt {4+2{\sqrt {2}}}}}}+{\sqrt {6-3{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12-6{\sqrt {2+{\sqrt {2}}}}}}+{\sqrt {12-6{\sqrt {2+{\sqrt {2}}}}+6{\sqrt {2}}-3{\sqrt {4+2{\sqrt {2}}}}}}}{4}}={\tfrac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09d240ffc9f8df6e820c866b30985968f58e2e71)
4.5°
![{\displaystyle {}_{\sin {\frac {\pi }{40}}=\sin 4.5^{\circ }={\tfrac {2{\sqrt {10+2{\sqrt {5}}+5{\sqrt {2}}+{\sqrt {1}}0}}-{\sqrt {20+4{\sqrt {5}}+10{\sqrt {2}}+2{\sqrt {10}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {20-10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c735897d1fb81d4d35d8d879dc636626e419d61)
![{\displaystyle {}_{\cos {\frac {\pi }{40}}=\cos 4.5^{\circ }={\tfrac {2{\sqrt {10+2{\sqrt {5}}-5{\sqrt {2}}-{\sqrt {1}}0}}+{\sqrt {20+4{\sqrt {5}}-10{\sqrt {2}}-2{\sqrt {10}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bae6c2d4bd7b3dc563bd2999a1a31d66eade277b)
7.5°
![{\displaystyle {}_{\sin {\frac {\pi }{24}}=\sin 7.5^{\circ }={\frac {{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {12+6{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17c95b10d28df67952e38699a988d68b8d41edbe)
![{\displaystyle {}_{\cos {\frac {\pi }{24}}=\cos 7.5^{\circ }={\tfrac {{\sqrt {6-3{\sqrt {2}}}}+{\sqrt {12-6{\sqrt {2}}}}+{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8649cd7ffb6c0c920f278ffb039e8a995978a139)
![{\displaystyle {}_{\tan {\frac {\pi }{24}}=\tan 7.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e229c173532d84509fdcaf112e94eb34e0f7e6)
![{\displaystyle {}_{\cot {\frac {\pi }{24}}=\cot 7.5^{\circ }=2+{\sqrt {2}}+{\sqrt {3}}+{\sqrt {6}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7333f1152e14da4896f4c9c9866826cfbe3a6c)
![{\displaystyle {}_{\csc {\frac {\pi }{24}}=\csc 7.5^{\circ }=4{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {24-6{\sqrt {2}}-6{\sqrt {6}}}}+5{\sqrt {4-{\sqrt {2}}-{\sqrt {6}}}}+3{\sqrt {12-3{\sqrt {2}}-3{\sqrt {6}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a50d6b3c12f3c0ffd5a3bbe05920626638c379)
![{\displaystyle {}_{\sec {\frac {\pi }{24}}=\sec 7.5^{\circ }=4{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {24+6{\sqrt {2}}+6{\sqrt {6}}}}-5{\sqrt {4+{\sqrt {2}}+{\sqrt {6}}}}-3{\sqrt {12+3{\sqrt {2}}+3{\sqrt {6}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee1aeabce8faf352e382ecec29a9e13252ba3b5)
11.25°
![{\displaystyle {}_{\sin {\frac {\pi }{16}}=\sin 11.25^{\circ }={\frac {\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d784711566ec82600b6df7c44ae2577b1624f845)
![{\displaystyle {}_{\cos {\frac {\pi }{16}}=\cos 11.25^{\circ }={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e2061a12210b635136ce8a7f318c37b093627e)
![{\displaystyle {}_{\tan {\frac {\pi }{16}}=\tan 11.25^{\circ }={\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2}}-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256b220fa608a02d477fb2111372d896ea152692)
![{\displaystyle {}_{\cot {\frac {\pi }{16}}=\cot 11.25^{\circ }={\sqrt {4+2{\sqrt {2}}}}+{\sqrt {2}}+1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e89180c19dd5bf7a0b774053e4b0fe7c750f5d47)
![{\displaystyle {}_{\csc {\frac {\pi }{16}}=\csc 11.25^{\circ }=4{\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4-2{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4+2{\sqrt {2}}-2{\sqrt {2+{\sqrt {2}}}}-{\sqrt {4+2{\sqrt {2}}}}}}+{\sqrt {8+4{\sqrt {2}}-4{\sqrt {2+{\sqrt {2}}}}-2{\sqrt {4+2{\sqrt {2}}}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d87444b74b2b66f062804c086dd2145f38202b1)
![{\displaystyle {}_{\sec {\frac {\pi }{16}}=\sec 11.25^{\circ }=4{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}+2{\sqrt {4+2{\sqrt {2+{\sqrt {2}}}}}}-2{\sqrt {4+2{\sqrt {2}}+2{\sqrt {2+{\sqrt {2}}}}+{\sqrt {4+2{\sqrt {2}}}}}}-{\sqrt {8+4{\sqrt {2}}+4{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {4+2{\sqrt {2}}}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/972b54e43b086114aa1c7739cf22692f409c64bc)
12.5°
![{\displaystyle {}_{\sin {\frac {5\pi }{72}}=\sin 12.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}+2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {3\pi }{40}}=\sin 13.5^{\circ }={\frac {{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efea59a98022dfcdb6535092bbbb40f56865fe42)
![{\displaystyle {}_{\cos {\frac {3\pi }{40}}=\cos 13.5^{\circ }={\frac {{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/401f218deb9e8b102d6ee80ab41b5a03464059f4)
17.5°
![{\displaystyle {}_{\sin {\frac {7\pi }{72}}=\sin 17.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {\pi }{8}}=\sin 22.5^{\circ }={\tfrac {\sqrt {2-{\sqrt {2}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f0835e18da004d5c073f7edd8feb5adb9aab2cf)
![{\displaystyle {}_{\cos {\frac {\pi }{8}}=\cos 22.5^{\circ }={\tfrac {\sqrt {2+{\sqrt {2}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc5df4a766445d5402e144c726e4ac65e6c176f)
![{\displaystyle {}_{\tan {\frac {\pi }{8}}=\tan 22.5^{\circ }={\sqrt {2}}-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/538842c835be7b013e7c6a66ad4ef4d54567f7fd)
![{\displaystyle {}_{\cot {\frac {\pi }{8}}=\cot 22.5^{\circ }={\sqrt {2}}+1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44f91b79cf0bd4261bd1aeca8c4cb6fc549d72d9)
![{\displaystyle {}_{\csc {\frac {\pi }{8}}=\csc 22.5^{\circ }={\sqrt {4+2{\sqrt {2}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af6a4253d93c6bd7d2afdac30e4e9fff545b5112)
![{\displaystyle {}_{\sec {\frac {\pi }{8}}=\sec 22.5^{\circ }={\sqrt {4-2{\sqrt {2}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2915d4ade4bc81b3c2fabfc95f9b4225a70c3a5)
27.5°
![{\displaystyle {}_{\sin {\frac {11\pi }{72}}=\sin 27.5^{\circ }={\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {7\pi }{40}}=\sin 31.5^{\circ }={\frac {{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1f501a5e3a75ed127d5859b34f3e9168401731)
![{\displaystyle {}_{\cos {\frac {7\pi }{40}}=\cos 31.5^{\circ }={\frac {{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1903fdd30591fa426a2feda9ba12e7590fd6c51b)
32.5°
![{\displaystyle {}_{\sin {\frac {13\pi }{72}}=\sin 32.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}+2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}+2{\sqrt {2}}}}-2{\sqrt {8-2{\sqrt {6}}-2{\sqrt {2}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {3\pi }{16}}=\sin 33.75^{\circ }={\tfrac {\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e384c17e47f5ef55d20f374eebb3eaf82f2506bf)
![{\displaystyle {}_{\cos {\frac {3\pi }{16}}=\cos 33.75^{\circ }={\tfrac {\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2708de260f70a2ada7249d9d362687b02a0d3d)
![{\displaystyle {}_{\tan {\frac {3\pi }{16}}=\tan 33.75^{\circ }={\sqrt {4-2{\sqrt {2}}}}-{\sqrt {2}}+1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5623b40a39c84221eaaf00c757f781cbe68bac67)
![{\displaystyle {}_{\cot {\frac {3\pi }{16}}=\cot 33.75^{\circ }={\sqrt {4-2{\sqrt {2}}}}+{\sqrt {2}}-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ee938e544149bff615c67a26cbc1eb89a2d0dc)
![{\displaystyle {}_{\csc {\frac {3\pi }{16}}=\csc 33.75^{\circ }=4{\sqrt {2-{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4-2{\sqrt {2-{\sqrt {2}}}}}}+2{\sqrt {4-2{\sqrt {2}}-2{\sqrt {2-{\sqrt {2}}}}+{\sqrt {4-2{\sqrt {2}}}}}}-{\sqrt {8-4{\sqrt {2}}-4{\sqrt {2-{\sqrt {2}}}}+2{\sqrt {4-2{\sqrt {2}}}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6712020f40338b65793bb6b1849eb0ed33ff6591)
![{\displaystyle {}_{\sec {\frac {3\pi }{16}}=\sec 33.75^{\circ }=4{\sqrt {2+{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4+2{\sqrt {2-{\sqrt {2}}}}}}-2{\sqrt {4-2{\sqrt {2}}+2{\sqrt {2-{\sqrt {2}}}}-{\sqrt {4-2{\sqrt {2}}}}}}+{\sqrt {8-4{\sqrt {2}}+4{\sqrt {2-{\sqrt {2}}}}-2{\sqrt {4-2{\sqrt {2}}}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dd766dcd7b7a970f1b12c0b1c0905b1aba2acba)
37.5°
![{\displaystyle {}_{\sin {\frac {5\pi }{24}}=\sin 37.5^{\circ }={\tfrac {{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2760a4bd643f27f23ad92039787ea2534deae93)
![{\displaystyle {}_{\cos {\frac {5\pi }{24}}=\cos 37.5^{\circ }={\tfrac {{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea95d1365ea808c0fabda3b85d1b69c58b175b0d)
![{\displaystyle {}_{\tan {\frac {5\pi }{24}}=\tan 37.5^{\circ }={\sqrt {3}}+{\sqrt {6}}-{\sqrt {2}}-2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c6a306c4a75ef94db2d508cd9c98efd4bd34fb8)
![{\displaystyle {}_{\cot {\frac {5\pi }{24}}=\cot 37.5^{\circ }=2+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3502111bf22d0cb35461d04f19cd2eb1f4aec466)
![{\displaystyle {}_{\csc {\frac {5\pi }{24}}=\csc 37.5^{\circ }=4{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}-2{\sqrt {24+6{\sqrt {2}}-6{\sqrt {6}}}}-5{\sqrt {4+{\sqrt {2}}-{\sqrt {6}}}}+3{\sqrt {12+3{\sqrt {2}}-3{\sqrt {6}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4df65dceb3c73c319e086daf4ea7a7e8dfddce)
![{\displaystyle {}_{\sec {\frac {5\pi }{24}}=\sec 37.5^{\circ }=4{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {24-6{\sqrt {2}}+6{\sqrt {6}}}}+5{\sqrt {4+{\sqrt {6}}-{\sqrt {2}}}}+3{\sqrt {12+3{\sqrt {6}}-3{\sqrt {2}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83e0455317647cd35f44d5696f05230abb70857b)
40.5°
![{\displaystyle {}_{\sin {\frac {9\pi }{40}}=\sin 40.5^{\circ }={\frac {{\sqrt {20-2{\sqrt {10}}+4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7af7152a7f1e0f870d460375e1eb1fd2fcbe72ad)
![{\displaystyle {}_{\cos {\frac {9\pi }{40}}=\cos 40.5^{\circ }={\frac {{\sqrt {20+2{\sqrt {10}}+4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e08a7e5ada53b2fd08a367a022aed5cbc85266dc)
![{\displaystyle {}_{\tan {\frac {9\pi }{40}}=\tan 40.5^{\circ }={\frac {3{\sqrt {18-2{\sqrt {5}}+8{\sqrt {5-{\sqrt {5}}}}}}-2{\sqrt {45-5{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}}}+{\sqrt {126+20{\sqrt {2}}+56{\sqrt {5-{\sqrt {5}}}}-14{\sqrt {5}}-36{\sqrt {10}}-16{\sqrt {50-10{\sqrt {5}}}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6f1efe50164ff5b4233cc82b07d1cee08bbd67)
![{\displaystyle {}_{\cot {\frac {9\pi }{40}}=\cot 40.5^{\circ }={\frac {13{\sqrt {18-2{\sqrt {5}}+8{\sqrt {5-{\sqrt {5}}}}}}+7{\sqrt {90-10{\sqrt {5}}+40{\sqrt {5-{\sqrt {5}}}}}}-16{\sqrt {25-7{\sqrt {5}}+10{\sqrt {5-{\sqrt {5}}}}-2{\sqrt {25-5{\sqrt {5}}}}}}+22{\sqrt {9-{\sqrt {5}}+4{\sqrt {5-{\sqrt {5}}}}}}+6{\sqrt {45-5{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}}}-12{\sqrt {50-14{\sqrt {5}}+20{\sqrt {5-{\sqrt {5}}}}-4{\sqrt {25-5{\sqrt {5}}}}}}-8{\sqrt {125-35{\sqrt {5}}+50{\sqrt {5-{\sqrt {5}}}}-10{\sqrt {25-5{\sqrt {5}}}}}}-4{\sqrt {250-70{\sqrt {5}}+100{\sqrt {5-{\sqrt {5}}}}-20{\sqrt {25-5{\sqrt {5}}}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/febdcfb4db09e2ef038c5f4ef8c11be9453815f5)
![{\displaystyle {}_{={\frac {{\sqrt {22+6{\sqrt {5}}-8{\sqrt {10+4{\sqrt {5}}}}}}+4{\sqrt {11+3{\sqrt {5}}-4{\sqrt {10+4{\sqrt {5}}}}}}+{\sqrt {110+30{\sqrt {5}}-40{\sqrt {10+4{\sqrt {5}}}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79f72fb9bd20f5a1852cce8510fae7493d97cf2c)
![{\displaystyle {}_{\sec {\frac {9\pi }{40}}=\sec 40.5^{\circ }={\frac {4{\sqrt {10+2{\sqrt {5}}-5{\sqrt {2}}-{\sqrt {10}}}}+{\sqrt {20+4{\sqrt {5}}-10{\sqrt {2}}-2{\sqrt {10}}}}-{\sqrt {100+20{\sqrt {5}}-50{\sqrt {2}}-10{\sqrt {10}}}}+2{\sqrt {20+10{\sqrt {2}}}}+6{\sqrt {4+2{\sqrt {2}}}}-2{\sqrt {10+5{\sqrt {2}}}}-10{\sqrt {2+{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7935e13d2210beff7a2ff14c3b20c05835482e4f)
42.5°
![{\displaystyle {}_{\sin {\frac {17\pi }{72}}=\sin 42.5^{\circ }={\frac {1-{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}+2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}+{\frac {1+{\sqrt {3}}{\rm {i}}}{8}}{\sqrt[{3}]{2{\sqrt {8+2{\sqrt {6}}-2{\sqrt {2}}}}-2{\sqrt {8+2{\sqrt {2}}-2{\sqrt {6}}}}{\rm {i}}}}}\,}](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°
![{\displaystyle {}_{\sin {\frac {11\pi }{40}}=\sin 49.5^{\circ }={\tfrac {{\sqrt {20+2{\sqrt {10}}+4{\sqrt {5}}+10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d107b7b65f4c5183e6fe1ec3806960646188106b)
![{\displaystyle {}_{\cos {\frac {11\pi }{40}}=\cos 49.5^{\circ }={\tfrac {{\sqrt {20-2{\sqrt {10}}+4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d808652e0f6469f5e75dbfb00d4d6fe27a565e)
58.5°
![{\displaystyle {}_{\sin {\frac {13\pi }{40}}=\sin 58.5^{\circ }={\tfrac {{\sqrt {10-5{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14bc20689b8ca65a2b81cd2f71229b8a52814d09)
![{\displaystyle {}_{\cos {\frac {13\pi }{40}}=\cos 58.5^{\circ }={\tfrac {{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}-{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}+10{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c539b9bb57e652375ca5e1ea0191d88a296f6e7)
67.5°
![{\displaystyle {}_{\sin {\frac {3\pi }{8}}=\sin 67.5^{\circ }={\tfrac {\sqrt {2+{\sqrt {2}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84599a21399e46b7f4b2859fd78a55b65e50ff04)
![{\displaystyle {}_{\cos {\frac {3\pi }{8}}=\cos 67.5^{\circ }={\tfrac {\sqrt {2-{\sqrt {2}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba9d821ad0b04aa288ea42704b6c40446e94a666)
![{\displaystyle {}_{\tan {\frac {3\pi }{8}}=\tan 67.5^{\circ }={\sqrt {2}}+1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c9aa2b7fcfc7362c3892dea3a5fa7545c15574)
![{\displaystyle {}_{\cot {\frac {3\pi }{8}}=\cot 67.5^{\circ }={\sqrt {2}}-1}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9977f430cc665ec126b7a1726d6e80154cb82ea4)
![{\displaystyle {}_{\csc {\frac {3\pi }{8}}=\csc 67.5^{\circ }={\sqrt {4-2{\sqrt {2}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe93cd74f385a462d6331287adc6af57159b260)
![{\displaystyle {}_{\sec {\frac {3\pi }{8}}=\sec 67.5^{\circ }={\sqrt {4+2{\sqrt {2}}}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b8f927a1eec9a1ebdf2750c2d883c3e70c50b2)
76.5°
![{\displaystyle {}_{\sin {\frac {17\pi }{40}}=\sin 76.5^{\circ }={\tfrac {{\sqrt {20+2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/702ce5806ab3801fdc531c24c450e61205b168b8)
![{\displaystyle {}_{\cos {\frac {17\pi }{40}}=\cos 76.5^{\circ }={\tfrac {{\sqrt {20-2{\sqrt {10}}-4{\sqrt {5}}-10{\sqrt {2}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3a44976dd1028af69a09666cd08a38d45a49e5e)
82.5°
![{\displaystyle {}_{\sin {\frac {11\pi }{24}}=\sin 82.5^{\circ }={\tfrac {{\sqrt {4+2{\sqrt {2}}}}-{\sqrt {2+{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}+{\sqrt {12-6{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb63a807434031a0f97933845900b043b12368fd)
![{\displaystyle {}_{\cos {\frac {11\pi }{24}}=\cos 82.5^{\circ }={\tfrac {{\sqrt {4-2{\sqrt {2}}}}+{\sqrt {2-{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {12+6{\sqrt {2}}}}}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f213697c9a780723367a46a1f1d610a00e4905e)
![{\displaystyle {}_{\tan {\frac {11\pi }{24}}=\tan 82.5^{\circ }={\sqrt {6}}+2+{\sqrt {5+2{\sqrt {6}}}}={\sqrt {3}}+{\sqrt {2}}+{\sqrt {6}}+2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d84635630f3867474c4d2fa8f6f2c8568d2e598)
![{\displaystyle {}_{\cot {\frac {11\pi }{24}}=\cot 82.5^{\circ }={\sqrt {6}}-2-{\sqrt {5-2{\sqrt {6}}}}={\sqrt {3}}-{\sqrt {2}}+{\sqrt {6}}-2}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10837a892daee0d6599c972068c2fa2045e79435)
cos(90/17)°
![{\displaystyle {}_{\cos {\pi \over 34}=\cos {\frac {90}{17}}^{\circ }={\tfrac {\sqrt {136+6{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}}}+8{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {289+51{\sqrt {17}}-17{\sqrt {34-2{\sqrt {17}}}}-34{\sqrt {34+2{\sqrt {17}}}}}}+4{\sqrt {476+60{\sqrt {17}}-34{\sqrt {34-2{\sqrt {17}}}}-68{\sqrt {34+2{\sqrt {17}}+2{\sqrt {578-34{\sqrt {17}}}}}}-2{\sqrt {578+34{\sqrt {17}}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c71f6a6bd673eafb41626c630e90200de876586b)
sin(360k/17)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}65536x^{16}-278528x^{14}+487424x^{12}-452608x^{10}+239360x^{8}-71808x^{6}+11424x^{4}-816x^{2}+17=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7334a604cf1dec6715ac801e1db3536836b4ecb)
![{\displaystyle {}_{\sin {2\pi \over 17}=\sin {\frac {360}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3399e4f711cdcf0d99352b0fcbaf164d475514e)
![{\displaystyle {}_{\sin {4\pi \over 17}=\sin {\frac {720}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22ab6d2a01a216ae8ba50e11c1aec43dd0bf2381)
![{\displaystyle {}_{\sin {6\pi \over 17}=\sin {\frac {1080}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8cc95cfbd91c277828dccf2c8280d1a4518b122)
![{\displaystyle {}_{\sin {8\pi \over 17}=\sin {\frac {1440}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd724ad0d9bc5d6628902070ebc9cb6d8429e6eb)
![{\displaystyle {}_{\sin {10\pi \over 17}=\sin {\frac {1800}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7aa4b6184d60573e259e2444731e583dbc1ad4c)
![{\displaystyle {}_{\sin {12\pi \over 17}=\sin {\frac {2160}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4864c99edf15c3880533eddc26b4f10f71f7b3)
![{\displaystyle {}_{\sin {14\pi \over 17}=\sin {\frac {2520}{17}}^{\circ }={\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2abc0edd0be46660f2db8aa5a569359421da343)
![{\displaystyle {}_{\sin {16\pi \over 17}=\sin {\frac {2880}{17}}^{\circ }={\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aebcbc4d248c0d5a2e41e4840d60577e690a1c03)
![{\displaystyle {}_{\sin {18\pi \over 17}=\sin {\frac {2880}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a1c4b9b39ace6c824bca6117bbbeecd64653a1)
![{\displaystyle {}_{\sin {20\pi \over 17}=\sin {\frac {3600}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0967aaa522369dc9213ebf79c0f3d3d6279b6224)
![{\displaystyle {}_{\sin {22\pi \over 17}=\sin {\frac {3960}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}-2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3130857319974ee148f61a50d558ebeafc8d8ab0)
![{\displaystyle {}_{\sin {24\pi \over 17}=\sin {\frac {4320}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/284a8f440d484fdbbbaa310ea49edad8eb8ab390)
![{\displaystyle {}_{\sin {26\pi \over 17}=\sin {\frac {4680}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}+4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cf1fdab72b858779c87d749b6f170713efed)
![{\displaystyle {}_{\sin {28\pi \over 17}=\sin {\frac {5040}{17}}^{\circ }=-{\frac {\sqrt {34+2{\sqrt {17}}+2{\sqrt {34+2{\sqrt {17}}}}-4{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc07f402673e7bb7433d140e3f30519d8c3a0278)
![{\displaystyle {}_{\sin {30\pi \over 17}=\sin {\frac {5400}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}-2{\sqrt {34-2{\sqrt {17}}}}+4{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7acc53e672ecff439e3c391add0562bf56df9cc1)
![{\displaystyle {}_{\sin {32\pi \over 17}=\sin {\frac {5760}{17}}^{\circ }=-{\frac {\sqrt {34-2{\sqrt {17}}+2{\sqrt {34-2{\sqrt {17}}}}-4{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}}{8}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ab6eb45d9c14d3ba57ad64b356d711ebdbfa50)
cos(360k/17)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}256x^{8}+128x^{7}-448x^{6}-192x^{5}+240x^{4}+80x^{3}-40x^{2}-8x+1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9055dd34cce7319900419580a4843f3b108dfb)
![{\displaystyle {}_{\cos {2\pi \over 17}=\cos {\frac {360}{17}}^{\circ }={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d91d072f15df70bdc6ae5cb0f757ea20cbf94c8)
![{\displaystyle {}_{\cos {4\pi \over 17}=\cos {\frac {720}{17}}^{\circ }={\frac {-1+{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0622a3c617728d485f891cbc815fac95f08e0990)
![{\displaystyle {}_{\cos {6\pi \over 17}=\cos {\frac {1080}{17}}^{\circ }={\frac {-1-{\sqrt {17}}+{\sqrt {34+2{\sqrt {17}}}}+2{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9af6c156d394dd29d2dda44c30dd4429d6f473b0)
![{\displaystyle {}_{\cos {8\pi \over 17}=\cos {\frac {1440}{17}}^{\circ }={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d164a9458669acfb1a0edcf8a6f0c70adcab4630)
![{\displaystyle {}_{\cos {10\pi \over 17}=\cos {\frac {1800}{17}}^{\circ }={\frac {-1-{\sqrt {17}}+{\sqrt {34+2{\sqrt {17}}}}-2{\sqrt {17-3{\sqrt {17}}+{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efb93dc0f0a4a82dbd577604693c494394f3fbf)
![{\displaystyle {}_{\cos {12\pi \over 17}=\cos {\frac {2160}{17}}^{\circ }={\frac {-1-{\sqrt {17}}-{\sqrt {34+2{\sqrt {17}}}}+2{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98afb22478e7c8764df6f65d7d9534dc0550b8a2)
![{\displaystyle {}_{\cos {14\pi \over 17}=\cos {\frac {2520}{17}}^{\circ }={\frac {-1-{\sqrt {17}}-{\sqrt {34+2{\sqrt {17}}}}-2{\sqrt {17-3{\sqrt {17}}-{\sqrt {170-38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bcf5ad10805171d0dfd2a4f35611ab3e6ebdf97)
![{\displaystyle {}_{\cos {16\pi \over 17}=\cos {\frac {2880}{17}}^{\circ }={\frac {-1+{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {17+3{\sqrt {17}}+{\sqrt {170+38{\sqrt {17}}}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/894da9d510fcf30e78efef495b32000030fbbe87)
sec(360k/17)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{8}-8x^{7}-40x^{6}+80x^{5}+240x^{4}-192x^{3}-448x^{2}+128x+256=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aaf0528ab7e866a466e7315208a09ff5ac857d7)
![{\displaystyle {}_{\sec {2\pi \over 17}=\sec {\frac {360}{17}}^{\circ }={\frac {2+{\sqrt {17}}+{\sqrt {17+4{\sqrt {17}}}}-{\sqrt {34+4{\sqrt {17}}+2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6812c7ccc7660ecf38e90b58c83a1bcbd42050bc)
![{\displaystyle {}_{\sec {4\pi \over 17}=\sec {\frac {720}{17}}^{\circ }={\frac {2+{\sqrt {17}}-{\sqrt {17+4{\sqrt {17}}}}+{\sqrt {34+4{\sqrt {17}}-2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36bd34754b4adabd83ac8e606ecc3203f20f6602)
![{\displaystyle {}_{\sec {6\pi \over 17}=\sec {\frac {1080}{17}}^{\circ }={\frac {2-{\sqrt {17}}+{\sqrt {17-4{\sqrt {17}}}}+{\sqrt {34-4{\sqrt {17}}+2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1906dbde467e8f1dfb1b98e24d1f66ea16d0bce)
![{\displaystyle {}_{\sec {8\pi \over 17}=\sec {\frac {1440}{17}}^{\circ }={\frac {2+{\sqrt {17}}+{\sqrt {17+4{\sqrt {17}}}}+{\sqrt {34+4{\sqrt {17}}+2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65e852c3166da741036ef43d3bc9a3f2db4911e8)
![{\displaystyle {}_{\sec {10\pi \over 17}=\sec {\frac {1800}{17}}^{\circ }={\frac {2-{\sqrt {17}}+{\sqrt {17-4{\sqrt {17}}}}-{\sqrt {34-4{\sqrt {17}}+2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea107d0687711e78a0d76a331db911b014c83035)
![{\displaystyle {}_{\sec {12\pi \over 17}=\sec {\frac {2160}{17}}^{\circ }={\frac {2-{\sqrt {17}}-{\sqrt {17-4{\sqrt {17}}}}-{\sqrt {34-4{\sqrt {17}}-2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2f0eb199228b8eb8806f824fbe64ec77280135)
![{\displaystyle {}_{\sec {14\pi \over 17}=\sec {\frac {2520}{17}}^{\circ }={\frac {2-{\sqrt {17}}-{\sqrt {17-4{\sqrt {17}}}}+{\sqrt {34-4{\sqrt {17}}-2{\sqrt {289-52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df5f22fe8a83bd9f790be30cbdac0ae47954eb82)
![{\displaystyle {}_{\sec {16\pi \over 17}=\sec {\frac {2880}{17}}^{\circ }={\frac {2+{\sqrt {17}}-{\sqrt {17+4{\sqrt {17}}}}-{\sqrt {34+4{\sqrt {17}}-2{\sqrt {289+52{\sqrt {17}}}}}}}{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a368c0faa73a16a3a0e1f079d0ad9a6f28da7f10)
csc(360k/17)°
![{\displaystyle {}_{\qquad {\mbox{Root of }}17x^{16}-816x^{14}+11424x^{12}-71808x^{10}+239360x^{8}-452608x^{6}+487424x^{4}-278528x^{2}+65536=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf40f54e887181e88a65d3072014008719db4bfd)
![{\displaystyle {}_{\csc {2\pi \over 17}=\csc {\frac {360}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85265ca9a635a36a893bbc99b9ac6703344dbd76)
![{\displaystyle {}_{\csc {4\pi \over 17}=\csc {\frac {720}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e69621f5e96e4be3dc05639bb33ee876929a0a1)
![{\displaystyle {}_{\csc {6\pi \over 17}=\csc {\frac {1080}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f37d6225097044784d8f32220be8995c7fc5c5fd)
![{\displaystyle {}_{\csc {8\pi \over 17}=\csc {\frac {1440}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45959e1663231ba7f560af5901d8d80a4c69a3f8)
![{\displaystyle {}_{\csc {10\pi \over 17}=\csc {\frac {1800}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da03afb673e90e47484510f14f61d10f595892d2)
![{\displaystyle {}_{\csc {12\pi \over 17}=\csc {\frac {2160}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa1b880a6a2d20e166f8ad7371becaf8c4554fd)
![{\displaystyle {}_{\csc {14\pi \over 17}=\csc {\frac {2520}{17}}^{\circ }={\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55abee1e54333c56f321e8253bf58e374d9189c0)
![{\displaystyle {}_{\csc {16\pi \over 17}=\csc {\frac {2880}{17}}^{\circ }={\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6ce1b5d0d7864cbbb28d1320eb4bac6fc838b7)
![{\displaystyle {}_{\csc {18\pi \over 17}=\csc {\frac {3240}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/321eb6c479c7621b00e573ef47ef7534fa83b75d)
![{\displaystyle {}_{\csc {20\pi \over 17}=\csc {\frac {3600}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/850ea9546f369d85236b7e24bca338f3e8974c9f)
![{\displaystyle {}_{\csc {22\pi \over 17}=\csc {\frac {3960}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}+289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}+34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83415967d5c5a14f892ce9d9d9697acec02cf7b9)
![{\displaystyle {}_{\csc {24\pi \over 17}=\csc {\frac {4320}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}-17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e226c69c130458ebdaad9c31c96b517ef043b428)
![{\displaystyle {}_{\csc {26\pi \over 17}=\csc {\frac {4680}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}-17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef515378bdbb37e0a013d7a3f4b87bb9de09e7e2)
![{\displaystyle {}_{\csc {28\pi \over 17}=\csc {\frac {5040}{17}}^{\circ }=-{\frac {\sqrt {1734-289{\sqrt {17}}-289{\sqrt {17-4{\sqrt {17}}}}+17{\sqrt {14450-3468{\sqrt {17}}-34{\sqrt {282353-68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6eb8b8de63713883bbb874772d2e694831532e)
![{\displaystyle {}_{\csc {30\pi \over 17}=\csc {\frac {5400}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}+289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}+34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38b733de017e8a2515cf15d9d695e31249d42db7)
![{\displaystyle {}_{\csc {32\pi \over 17}=\csc {\frac {5760}{17}}^{\circ }=-{\frac {\sqrt {1734+289{\sqrt {17}}-289{\sqrt {17+4{\sqrt {17}}}}+17{\sqrt {14450+3468{\sqrt {17}}-34{\sqrt {282353+68476{\sqrt {17}}}}}}}}{17}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8b7ed2cc958d55f7605ffe266639a489eee9c3)
tan7.5°、tan37.5°、tan52.5°、tan82.5°
![{\displaystyle {}_{\tan {\frac {\pi }{24}}=\tan 7.5^{\circ }={\sqrt {6}}+{\sqrt {2}}-2-{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26e229c173532d84509fdcaf112e94eb34e0f7e6)
![{\displaystyle {}_{\tan {\frac {5\pi }{24}}=\tan 37.5^{\circ }={\sqrt {6}}+{\sqrt {3}}-2-{\sqrt {2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad376b329d6331b269bb020b372f24b25ce2d028)
![{\displaystyle {}_{\tan {\frac {7\pi }{24}}=\tan 52.5^{\circ }=2+{\sqrt {6}}-{\sqrt {2}}-{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79d9d03e281efc18648a33e84528cc205ccc7045)
![{\displaystyle {}_{\tan {\frac {11\pi }{12}}=\tan 82.5^{\circ }=2+{\sqrt {6}}+{\sqrt {2}}+{\sqrt {3}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd24801a091eb92951e39d4d60c3663c16991376)
sin(360k/7)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}64x^{6}-112x^{4}+56x^{2}-7=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d5207a497a3ee05fa458083e434f691513a9ea)
![{\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)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}8x^{3}+4x^{2}-4x-1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0712f70432da6540b985bf813fafae9c3fe3eaf)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{6}-21x^{4}+35x^{2}-7=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71a74530274b1fbde597c1f943b9d0f7be4d8d18)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}7x^{6}-35x^{4}+21x^{2}-1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ba8014d35d2def817a3a348ef817ad8035428e)
![{\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)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{3}+4x^{2}-4x-8=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b23c22911dab9c905ca31921b314c1b3d2638180)
![{\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)°
![{\displaystyle {}_{\qquad {\mbox{Roots of }}7x^{6}-56x^{4}+112x^{2}-64=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8cba00c352ca383f94a3968d87e3703e0bdfa3e)
![{\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
![{\displaystyle {}_{\qquad {\mbox{Roots of }}1024x^{10}-2816x^{8}+2816x^{6}-1232x^{4}+220x^{2}-11=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f809aee8df61bba83a5b6a1227d41a17ad56bc)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}32x^{5}+16x^{4}-32x^{3}-12x^{2}+6x+1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e9ad2202d09d29acf48ce5ac5d7bfb80bc67a2)
![{\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)
註:該五次方程的預解方程及其根
![{\displaystyle {}_{\qquad {\mbox{Roots of }}z^{4}+{\frac {979}{32}}z^{3}+{\frac {467181}{1024}}z^{2}+{\frac {157668929}{32768}}z+{\frac {25937424601}{1048576}}=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/442f7e406c586a92ee58db169857fcf103f02bed)
![{\displaystyle {}_{z_{1}={\frac {-979+275{\sqrt {5}}+55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a414e16af29e1412cebda8458fb7b6e29dafe04)
![{\displaystyle {}_{z_{2}={\frac {-979+275{\sqrt {5}}-55{\sqrt {410+178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a1d7ed96f2ec90f2b631edd48cb78877a6caa)
![{\displaystyle {}_{z_{3}={\frac {-979-275{\sqrt {5}}+55{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4ecb901c1e1901fbac00d3f7a43811eab3afb3e)
![{\displaystyle {}_{z_{4}={\frac {-979-275{\sqrt {5}}-55{\sqrt {410-178{\sqrt {5}}}}{\rm {i}}}{128}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/152953a6d41f8d73e92014265c15d4a885a40219)
tan(360°k/11)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{10}-55x^{8}+330x^{6}-462x^{4}+165x^{2}-11=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69eac9214c2464d2c21c33522f2a818c12db21dc)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{5}+6x^{4}-12x^{3}-32x^{2}+16x+32=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33f85c30aca916d0757be0ae7595ddc4442cc86)
![{\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
![{\displaystyle {}_{\qquad {\mbox{Roots of }}4096x^{12}-13312x^{10}+16640x^{8}-9984x^{6}+2912x^{4}-364x^{2}+13=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a9daac2d787badf22b42fb7d673cc3b5b46f8c)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}64x^{6}+32x^{5}-80x^{4}-32x^{3}+24x^{2}+6x-1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3796fe686700b90f0bf412ac52e02201abe07ea3)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}13x^{12}-364x^{10}+2912x^{8}-9984x^{6}+16640x^{4}-13312x^{2}+4096=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32f6ba523c3cc514099da9ec2d3e19992c6a0af5)
![{\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
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{6}-6x^{5}-24x^{4}+32x^{3}+80x^{2}-32x-64=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/361b9d98327501ad90d3f3853a2440a6bcd4ac31)
![{\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
![{\displaystyle {}_{\qquad {\mbox{Roots of }}64x^{6}-32x^{5}-96x^{4}+48x^{3}+32x^{2}-16x+1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21178dd9763edf8bad3013949c65c4ec41857a6a)
![{\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
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{6}-16x^{5}+32x^{4}+48x^{3}-96x^{2}-32x+64=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e30f2ca159c75703d688650528c98471d92fe2d)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}512x^{9}+256x^{8}-1024x^{7}-448x^{6}+672x^{5}+240x^{4}-160x^{3}-40x^{2}+10x+1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08ea62627d4c11bdf75e601b8844a43c8e2a7133)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Root of }}2048x^{11}+1024x^{10}-5120x^{9}-2304x^{8}+4608x^{7}+1792x^{6}-1792x^{5}-560x^{4}+280x^{3}+60x^{2}-12x-1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c6be4d602941b1898b3981d537176dd070d2bf)
![{\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)
![{\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/16273141dafd24a90a0404b07395a084e9637321)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}x^{9}+18x^{8}-240x^{6}+864x^{4}-1152x^{2}+512=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3abe3eb5f7d7c897b0980017803b59cea69ad691)
![{\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)
![{\displaystyle {}_{\qquad {\mbox{Roots of }}4294967296x^{32}-34359738368x^{30}+124554051584x^{28}-270582939648x^{26}+392603631616x^{24}-401411670016x^{22}+297388736512x^{20}-161694613504x^{18}+64647462912x^{16}-18868600832x^{14}+3953983488x^{12}-578486272x^{10}+56539136x^{8}-3436544x^{6}+114176x^{4}-1536x^{2}+1=0}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ff4166e2118063167b322874c735285a47ee6b)
![{\displaystyle {}_{\sin {\frac {\pi }{120}}=\sin 1.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5016e8c8f3c63d6f6b49563c58f13ea073e39db)
![{\displaystyle {}_{\sin {\frac {7\pi }{120}}=\sin 10.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}-{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}+4{\sqrt {5}}+2{\sqrt {10}}}}+{\sqrt {60-6{\sqrt {10}}-30{\sqrt {2}}+12{\sqrt {5}}}}+{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c9cdb3f8081af876c78d7e4df4834652300bf8)
![{\displaystyle {}_{\sin {\frac {11\pi }{120}}=\sin 16.5^{\circ }={\frac {{\sqrt {30-15{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}+2{\sqrt {10-2{\sqrt {5}}+5{\sqrt {2}}-{\sqrt {10}}}}-2{\sqrt {30-6{\sqrt {5}}-15{\sqrt {2}}+3{\sqrt {10}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00b4c1a3c332fe63d32260a8f51e8cda1383d473)
![{\displaystyle {}_{\sin {\frac {59\pi }{120}}=\sin 88.5^{\circ }={\frac {{\sqrt {30-15{\sqrt {2}}}}+{\sqrt {6-3{\sqrt {2}}}}-{\sqrt {20-10{\sqrt {2}}-4{\sqrt {5}}+2{\sqrt {10}}}}+{\sqrt {60-6{\sqrt {10}}+30{\sqrt {2}}-12{\sqrt {5}}}}+{\sqrt {10+5{\sqrt {2}}}}+{\sqrt {2+{\sqrt {2}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/682a9d43a458663a8e83a228ffe4c5d4897b17e8)
![{\displaystyle {}_{\sin {\frac {119\pi }{120}}=\sin 178.5^{\circ }={\frac {{\sqrt {30+15{\sqrt {2}}}}+{\sqrt {6+3{\sqrt {2}}}}-{\sqrt {20+10{\sqrt {2}}-4{\sqrt {5}}-2{\sqrt {10}}}}-{\sqrt {60+6{\sqrt {10}}-30{\sqrt {2}}-12{\sqrt {5}}}}-{\sqrt {10-5{\sqrt {2}}}}-{\sqrt {2-{\sqrt {2}}}}}{16}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea6df09c9a27f778f25faac8e5dc86fe5cfcb42)
tan[k(arctan4)]
![{\displaystyle {}_{\tan {\frac {\arctan 4}{4}}={\frac {{\sqrt {34+2{\sqrt {17}}}}-{\sqrt {17}}-1}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96c7822e9b7d68d5f3e16f20bb9ab9cadef2bd08)
![{\displaystyle {}_{\tan {\frac {\arctan 4}{2}}={\frac {{\sqrt {17}}-1}{4}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83f07b3ff4f64607faeac383c05573cf37676b5)
![{\displaystyle {}_{\tan {\frac {3}{4}}\arctan 4={\frac {17{\sqrt {17}}-47+{\sqrt {9826-1598{\sqrt {17}}}}}{52}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c299331015629f9283ba827b86b7b0040141aef)
![{\displaystyle {}_{\tan {\frac {5}{4}}\arctan 4=-{\frac {1121+289{\sqrt {17}}+{\sqrt {2839714+647938{\sqrt {17}}}}}{404}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ed03b9d1cd2dac10054c382dbe80ad85200640)
![{\displaystyle {}_{\tan {\frac {3}{2}}\arctan 4=-{\frac {47+17{\sqrt {17}}}{52}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bacf7a7abf3ccd6b94da2fb4a9a5e928e9281572)
![{\displaystyle {}_{\tan {\frac {7}{4}}\arctan 4=-{\frac {20047+4913{\sqrt {17}}+17{\sqrt {2839714+681598{\sqrt {17}}}}}{2908}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc7fefce8cda7e1684913330f51267483e602077)
![{\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)