8 π 3 ⇒ ( − 1 2 , 3 2 ) {\displaystyle {\frac {8\pi }{3}}\Rightarrow \left(-{\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)}
sin ( 8 π 3 ) = 3 2 csc ( 8 π 3 ) = 2 1 = 2 cos ( 8 π 3 ) = 2 2 sec ( 8 π 3 ) = 2 − 3 ⋅ 3 3 = − 2 3 3 tan ( 8 π 3 ) = 1 2 − 3 2 = ( 1 2 ) ( − 2 3 ) = − 1 3 ⋅ 3 3 = − 3 3 cot ( 8 π 3 ) = − 3 1 = − 3 {\displaystyle {\begin{aligned}\sin {\left({\frac {8\pi }{3}}\right)}&={\frac {\sqrt {3}}{2}}&\csc {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{1}}=2\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{2}}&\sec {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{-{\sqrt {3}}}}\cdot {\frac {\sqrt {3}}{\sqrt {3}}}=-{\frac {2{\sqrt {3}}}{3}}\\[2ex]\tan {\left({\frac {8\pi }{3}}\right)}&={\frac {\frac {1}{2}}{-{\frac {\sqrt {3}}{2}}}}=\left({\frac {1}{2}}\right)\left(-{\frac {2}{\sqrt {3}}}\right)=-{\frac {1}{\sqrt {3}}}\cdot {\frac {\sqrt {3}}{\sqrt {3}}}=-{\frac {\sqrt {3}}{3}}&\cot {\left({\frac {8\pi }{3}}\right)}&=-{\frac {\sqrt {3}}{1}}=-{\sqrt {3}}\\[2ex]\end{aligned}}}
![{\displaystyle {\begin{aligned}\sin {\left({\frac {8\pi }{3}}\right)}&={\frac {\sqrt {3}}{2}}&\csc {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{1}}=2\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{2}}&\sec {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{-{\sqrt {3}}}}\cdot {\frac {\sqrt {3}}{\sqrt {3}}}=-{\frac {2{\sqrt {3}}}{3}}\\[2ex]\tan {\left({\frac {8\pi }{3}}\right)}&={\frac {\frac {1}{2}}{-{\frac {\sqrt {3}}{2}}}}=\left({\frac {1}{2}}\right)\left(-{\frac {2}{\sqrt {3}}}\right)=-{\frac {1}{\sqrt {3}}}\cdot {\frac {\sqrt {3}}{\sqrt {3}}}=-{\frac {\sqrt {3}}{3}}&\cot {\left({\frac {8\pi }{3}}\right)}&=-{\frac {\sqrt {3}}{1}}=-{\sqrt {3}}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/0b254372ac07c206318cd49be5028a3a9da672f1)