Question

Sebe-se que o
$\sin( \alpha ) = \frac{ \sqrt{2} }{3} \: e \: que \: 0 \leqslant \alpha \leqslant \frac{\pi}{2} \: determine \: a) \cos( \alpha ) \: e \: b) \tan( \alpha )$
Por favor URGENTE​

Answer 1

Vamos usar a relação trigonométrica fundamental:

$\sin^{2}\alpha+\cos^2 \alpha=1\\\\\bigg( \dfrac{\sqrt{2} }{3} \bigg) ^2+\cos^2 \alpha=1\\\\ \dfrac{2}{9}+\cos^2 \alpha=1\\\\\cos^2 \alpha=1- \dfrac{2}{9}\\\\\cos^2 \alpha=\dfrac{7}{9} \\\\\cos \alpha=\dfrac{\sqrt{7}}{3}$

$\tan \alpha =\dfrac{\sin \alpha }{\cos \alpha } \\\\\tan \alpha =\dfrac{\frac{\sqrt{2} }{3} }{\frac{\sqrt{7} }{3} } \\\\\tan \alpha =\dfrac{\sqrt{2} }{3} \cdot\dfrac{3}{\sqrt{7}} \\\\\tan \alpha =\dfrac{\sqrt{2} }{\sqrt{7}} \cdot\dfrac{\sqrt{7}}{\sqrt{7}} \\\\\tan \alpha =\dfrac{\sqrt{14} }{7}$