Question

Calcule o valor de cosx em cos x . sen x= raiz de 2/3 e tgx= raiz de 2. Dado 0 < x < pi e cos x > 0

Answer 1

Queremos o valor de cos(x) em :

$\displaystyle \text{cos(x)sen(x)}=\frac{\sqrt 2}{3} \ \ ; \ \ \text{tg(x)}= \sqrt2 \\\\\ \underline{\text{abrindo a tangente }}: \\\\ \text{tg(x)}=\sqrt{2}\to \frac{\text{sen( x)}}{\text{cos(x)}} = \sqrt{2} \\\\\ \text{sen(x)} = \text{cos(x)}.\sqrt{2} \\\\ \underline{\text{substituindo em}}:\\\\ \text{cos(x)sen(x)}=\frac{\sqrt2}{3} \\\\ \text{cos(x).cos(x)}\sqrt{2}=\frac{\sqrt 2}{3}\\\\ \text{cos}^2(\text x) = \frac{\sqrt{2}}{\sqrt{2}.3} \\\\\\ \text{cos(x)} = \frac{1}{\sqrt{3}}$

Portanto :

$\huge\boxed{\displaystyle \text{cos(x)} =\frac{\sqrt{3}}{3} \ }\checkmark$