6.2 Trigonometric Functions: Unit Circle Approach/78

$\theta \rightarrow (5,-12)$

$\theta \rightarrow x=5,\,y=-12,\,r=\,?$

${\begin{aligned}(5)^{2}+(-12)^{2}&=r^{2}\\25+144&=r^{2}\\169&=r^{2}\\{\sqrt {169}}&={\sqrt {r^{2}}}\\13&=r\\\end{aligned}}$

$\theta \rightarrow x=5,\,y=-12,\,r=13$

${\begin{aligned}\sin {(\theta )}&={\frac {-12}{13}}&\csc {(\theta )}&={\frac {2}{1}}=2\\[2ex]\cos {(\theta )}&={\frac {-{\sqrt {3}}}{2}}&\sec {(\theta )}&=3\\[2ex]\tan {(\theta )}&={\frac {2}{3}}&\cot {(\theta )}&=3\\[2ex]\end{aligned}}$