6.2 Trigonometric Functions: Unit Circle Approach/53

${\frac {8\pi }{3}}\Rightarrow \left(-{\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$

${\begin{aligned}\sin {\left({\frac {8\pi }{3}}\right)}&={\frac {\sqrt {3}}{2}}&\csc {\left({\frac {8\pi }{3}}\right)}&={{\frac {2}{\sqrt {3}}}\cdot {\sqrt {3}}}={\frac {2{\sqrt {3}}}{3}}\\[2ex]\cos {\left({\frac {8\pi }{3}}\right)}&=-{\frac {1}{2}}&\sec {\left({\frac {8\pi }{3}}\right)}&={\frac {2}{-{\sqrt {3}}}}\cdot {\frac {\sqrt {3}}{\sqrt {3}}}=-{\frac {2{\sqrt {3}}}{3}}\\[2ex]\tan {\left({\frac {8\pi }{3}}\right)}&={\frac {\frac {\sqrt {3}}{2}}{-{\frac {1}{2}}}}=\left(2\right)={\frac {\sqrt {3}}{-1}}=-{\sqrt {3}}&\cot {\left({\frac {8\pi }{3}}\right)}&=-{\frac {\sqrt {3}}{1}}=-{\sqrt {3}}\\[2ex]\end{aligned}}$