Question

sabendo-se que a cotgn= -2 e que x e ao 4* quadrante, assinale o item referente ao valor de cossec x

Answer 1

$\cot{x}=-2\\\\ \dfrac{3\pi}{2}\leq x \leq2\pi\ \therefore\ -1\leq\sin{x}\leq0\ \therefore\ \csc{x}<0$

$\boxed{\sin^2{x}+\cos^2{x}=1}\ \therefore\ \dfrac{\sin^2{x}}{\sin^2{x}}+\dfrac{\cos^2{x}}{\sin^2{x}}=\dfrac{1}{\sin^2{x}}\ \therefore$

$\boxed{1+\cot^2{x}=\csc^2{x}}$

$\text{Portanto:}\\\\ 1+(-2)^2=\csc^2{x}\ \therefore\ 5=\csc^2{x}\ \therefore\ \csc{x}=\pm\sqrt{5}\\\\ \mathrm{Como}\ \csc{x}<0\text{, teremos que:}\\\\ \boxed{\csc{x}=-\sqrt{5}}$