Question

√2 + 6
determine X e y na
4
Dado sen 750
figura abaixo
Y
-Z​

Answer 1

Usando Lei dos senos  :

$\displaystyle \frac{\text y }{\text{sen}(45^\circ) } = \frac{6}{\text{sen}(30^\circ)} = \frac{\text x}{\text{sen(X}^\circ)}$

$\displaystyle \frac{\text y }{\frac{\sqrt2}{2} } = \frac{6}{\frac{1}{2}} \\\\\\ \text y = \frac{\sqrt{2}}{2}.6.\frac{2}{1} \\\\\\ \huge\boxed{\text y = 6\sqrt{2}}\checkmark$

O ângulo em X. Sabendo que a soma dos ângulos internos de uma triângulo vale 180º, temos :

$\text X^\circ + 45^\circ +30^\circ = 180^\circ \\\\ \text X^\circ = 180^\circ-45^\circ-30^\circ \\\\ \text X^\circ = 180^\circ - 75^\circ$

Voltando para Lei dos Senos :

$\displaystyle \frac{6}{\text{sen}(30^\circ)} = \frac{\text x}{\text{sen(X}^\circ)} \\\\\\ \frac{\text x}{\text{sen}(180^\circ-75^\circ)} = \frac{6}{\text{sen(30}^\circ)}$

Sabemos que :

$\text{sen(}180^\circ-\theta) = \text{sen}(\theta)$

Portanto :

$\displaystyle \frac{\text x}{\text{sen}(180^\circ-75^\circ)} = \frac{6}{\text{sen(30}^\circ)} \\\\\\ \frac{\text x}{\text{sen}(75^\circ)} = \frac{6}{\text{sen(30}^\circ)} \\\\\\ \text x = \text{sen}(75^\circ).\frac{6}{\text{sen}(30^\circ)} \\\\\\ \text x = \frac{(\sqrt{2}+\sqrt{6})}{4}.\frac{6}{\frac{1}{2}} \\\\\\ \text x = \frac{(\sqrt{2}+\sqrt{6})}{4}.6.2 \\\\\\ \huge\boxed{\text x = 3(\sqrt{2}+\sqrt{6})}\checkmark$