6.2 Trigonometric Functions: Unit Circle Approach/53

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{8\pi}{3}\Rightarrow \left(-\frac{1}{2} ,\frac{\sqrt{3}}{2}\right)}

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