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 ) = 1 3 2 ⋅ 2 = 2 3 ⋅ 3 = 2 3 3 cos ( 8 π 3 ) = − 1 2 sec ( 8 π 3 ) = 1 − 1 2 ⋅ 2 = − 2 1 = − 2 tan ( 8 π 3 ) = 3 2 − 1 2 ⋅ 2 = 3 − 1 = − 3 cot ( 8 π 3 ) = − 1 2 3 2 ⋅ 2 = − 1 3 ⋅ 3 = − 3 3 {\displaystyle {\begin{aligned}\sin {\left({\frac {8\pi }{3}}\right)}&={\frac {\sqrt {3}}{2}}&\csc {\left({\frac {8\pi }{3}}\right)}&={\frac {1}{\frac {\sqrt {3}}{\cancel {2}}}}\cdot {\cancel {2}}={\frac {2}{\sqrt {3}}}\cdot {\sqrt {3}}={\frac {2{\sqrt {3}}}{3}}\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&=-{\frac {1}{2}}&\sec {\left({\frac {8\pi }{3}}\right)}&={\frac {1}{-{\frac {1}{\cancel {2}}}}}\cdot {\cancel {2}}=-{\frac {2}{1}}=-2\\[2ex]\tan {\left({\frac {8\pi }{3}}\right)}&={\frac {\frac {\sqrt {3}}{\cancel {2}}}{-{\frac {1}{\cancel {2}}}}}\cdot {\cancel {2}}={\frac {\sqrt {3}}{-1}}=-{\sqrt {3}}&\cot {\left({\frac {8\pi }{3}}\right)}&={\frac {-{\frac {1}{\cancel {2}}}}{\frac {\sqrt {3}}{\cancel {2}}}}\cdot {\cancel {2}}={\frac {-1}{\sqrt {3}}}\cdot {\sqrt {3}}={\frac {-{\sqrt {3}}}{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 {1}{\frac {\sqrt {3}}{\cancel {2}}}}\cdot {\cancel {2}}={\frac {2}{\sqrt {3}}}\cdot {\sqrt {3}}={\frac {2{\sqrt {3}}}{3}}\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&=-{\frac {1}{2}}&\sec {\left({\frac {8\pi }{3}}\right)}&={\frac {1}{-{\frac {1}{\cancel {2}}}}}\cdot {\cancel {2}}=-{\frac {2}{1}}=-2\\[2ex]\tan {\left({\frac {8\pi }{3}}\right)}&={\frac {\frac {\sqrt {3}}{\cancel {2}}}{-{\frac {1}{\cancel {2}}}}}\cdot {\cancel {2}}={\frac {\sqrt {3}}{-1}}=-{\sqrt {3}}&\cot {\left({\frac {8\pi }{3}}\right)}&={\frac {-{\frac {1}{\cancel {2}}}}{\frac {\sqrt {3}}{\cancel {2}}}}\cdot {\cancel {2}}={\frac {-1}{\sqrt {3}}}\cdot {\sqrt {3}}={\frac {-{\sqrt {3}}}{3}}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/e84bee6192feaf91f5efed7e735f9ff95013fdeb)