6.2 Trigonometric Functions: Unit Circle Approach/49

$210\Rightarrow ({\frac {7\pi }{6}})\Rightarrow ({\frac {-{\sqrt {3}}}{2}},{\frac {-1}{2}})$

${\begin{aligned}\sin {\left({\frac {7\pi }{6}}\right)}&={\frac {-1}{2}}&\csc {\left({\frac {7\pi }{6}}\right)}&={\frac {1}{\frac {-1}{2}}}\cdot {2}={\frac {2}{-1}}=-2\\[2ex]\cos {\left({\frac {7\pi }{6}}\right)}&={\frac {-{\sqrt {3}}}{2}}&\sec {\left({\frac {7\pi }{6}}\right)}&={\frac {1}{\frac {-{\sqrt {3}}}{2}}}\cdot {2}={\frac {2}{-{\sqrt {3}}}}\cdot {-{\sqrt {3}}}={\frac {2-{\sqrt {3}}}{3}}\\[2ex]\tan {\left({\frac {7\pi }{6}}\right)}&={\frac {\frac {-{\sqrt {3}}}{2}}{\frac {-1}{2}}}\cdot {2}=-{\frac {\sqrt {3}}{1}}=-{\sqrt {3}}&\cot {\left({\frac {7\pi }{6}}\right)}&={\frac {\frac {-{\sqrt {3}}}{2}}{\frac {-1}{2}}}\cdot {2}=-{\frac {\sqrt {3}}{1}}\\[2ex]\end{aligned}}$