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 3 ⋅ 3 = cos ( 8 π 3 ) = − 1 2 sec ( 8 π 3 ) = 2 − 3 ⋅ 3 3 = − 2 3 3 tan ( 8 π 3 ) = 3 2 − 1 2 = ( 2 ) = 3 − 1 = − 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}{{\sqrt {3}}\cdot {\sqrt {3}}}}=\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&=-{\frac {1}{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 {\sqrt {3}}{2}}{-{\frac {1}{2}}}}=\left(2\right)={\frac {\sqrt {3}}{-1}}=-{\sqrt {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}{{\sqrt {3}}\cdot {\sqrt {3}}}}=\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&=-{\frac {1}{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 {\sqrt {3}}{2}}{-{\frac {1}{2}}}}=\left(2\right)={\frac {\sqrt {3}}{-1}}=-{\sqrt {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/b374ce949c3a97d9c0ee0ce9b207285becd10b08)