Question

Fie x apartine (pi/2, pi) si cos2x=-1/2.Se cer cos x si sin x.

Answer 1

$x \: \in \: ( \frac{\pi}{2} ,\pi) = cadranul \: II = > sin \: x > 0,cos \: x < 0$

$cos2x = - \frac{1}{2}$

$cos2x = {cos}^{2} x - {sin}^{2} x$

$\left\{\begin{matrix}</p><p> {cos}^{2}x - {sin}^{2}x = - \frac{1}{2} \\ {sin}^{2} x + {cos}^{2} x = 1</p><p></p><p>\end{matrix}\right.$

${cos}^{2} x - {sin}^{2} x + {sin}^{2} x + {cos}^{2} x = - \frac{1}{2} + 1$

$2 {cos}^{2} x = \frac{1}{2}$

${cos}^{2} x = \frac{1}{4}$

$cos \: x = \pm \: \sqrt{ \frac{1}{4} } = \pm \: \frac{1}{2}$

$x \: \in \: ( \frac{\pi}{2} ,\pi) = > cos \: x = - \frac{1}{2}$

${sin}^{2} x + {cos}^{2} x = 1$

${sin}^{2} x + \frac{1}{4} = 1$

${sin}^{2} x = 1 - \frac{1}{4}$

${sin}^{2} x = \frac{3}{4}$

$sin \: x = \pm \: \sqrt{ \frac{3}{4} } = \pm \: \frac{ \sqrt{3} }{ \sqrt{4} } = \pm \: \frac{ \sqrt{3} }{2}$

$x \: \in \: ( \frac{\pi}{2} ,\pi) = > sin \: x = \frac{ \sqrt{3} }{2}$