Question

Sabendo que tg x = 2/3 raiz de 13 , determine o valor numérico da expressão cossec x+cos x /sec x +sen x

Answer 1

Vou admitir que o arco pertence ao primeiro quadrante.

$\cot(x) = \frac{3 \sqrt{13} }{2} \\ { \cot }^{2}(x) = {(\frac{3 \sqrt{13} }{2}) }^{2} = \frac{117}{4}$

$\csc(x) = \sqrt{1 + { \cot}^{2}(x)} \\ \csc(x) = \sqrt{1 + \frac{117}{4} } \\ = \sqrt{ \frac{121}{4} }=\frac{11}{2}$

$\sin(x) = \frac{2}{11}$

$\cos(x) = \frac{\cancel2}{11}. \frac{3 \sqrt{13} }{\cancel2} = \frac{3 \sqrt{13} }{11}$

$\sec(x) = \frac{11}{3 \sqrt{13} } = \frac{11 \sqrt{13} }{39}$

$\frac{ \csc(x) + \cos(x) }{ \sec(x) + \sin(x)} \\ = \frac{ \frac{11}{2} + \frac{3 \sqrt{13} }{11}}{ \frac{11 \sqrt{13} }{39} + \frac{2}{11} }$

$( \frac{11}{2} + \frac{3 \sqrt{13} }{11}) \div ( \frac{11 \sqrt{13} }{39} + \frac{2}{11}) \\ (\frac{121 + 3 \sqrt{13}}{22}) \div ( \frac{121 \sqrt{13} + 78}{429} )$

$\frac{(121 + 3 \sqrt{13})}{ \cancel{22} \: 11}. \frac{ \cancel{429} \: 39}{(121 \sqrt{13} + 78) } \\ = \frac{4719 + 117 \sqrt{13} }{1331 \sqrt{13} + 858}$