6.2 Trigonometric Functions: Unit Circle Approach/47

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

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