Question

 Sabendo-se que sen x = $\frac{2}{5}$, com $\frac{\pi}{2}$ <x<$\pi$, determine:
a) sen 2 x
b) cos 2 x

Answer 1

$sen\ 2x = 2\ senx.cosx\\\\\ sen^2x+cos^2x = 1\\\\\ cos^2x = 1-\frac{4}{9}\\\\\ cos^2x = \frac{5}{4}\\\\ cos\ x = -\frac{ \sqrt{5} }{2}$

$sen\ 2x = 2.\ \frac{2}{5}.-\frac{ \sqrt{5} }{2}\\\\\ \boxed{sen\ 2x = -2\frac{\sqrt{5}}{5}}}$

$cos\ 2x = 1-cos^2x\\\\ cos\ 2x = 1-(-\frac{\sqrt{5}}{2})\\\\ cos\ 2x = 1+\frac{\sqrt{5}}{2}\\\\ \boxed{cos\ 2x = \frac{2+\sqrt{5}}{2}}$