6.2 Trigonometric Functions: Unit Circle Approach/53

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

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