Question

Resolva o sistema
$\left \{ {{cos(x + y) = -1} \atop {x - y =\frac{\pi }{2} }} \right.$

Answer 1

$\displaystyle \left \{ {{ \text i) \ \displaystyle \text{cos(x+y)}=1} \atop {\text{ii}) \ \displaystyle \text x-\text y=\frac{\pi}{2}}} \right.$

Trabalhando na 1ª equação :

$\text{cos(x+y)}= 1 \\\\\ \text{x+y} = 0+2\text k.\pi \\\\ \text{x+y} = 2\text k.\pi \ , \ \underline{\text{com k}\in \mathbb{Z}}$

Então, temos :

$\displaysytle \text i) \ \text{x+y} = 2.\text k.\pi$

$\displaystyle \text{ii}) \ \text x-\text y = \frac{\pi}{2}$  

somando as equação :

$\displaystyle \text x + \text y + \text x-\text y = 2.\text k.\pi + \frac{\pi}{2} \\\\ 2.\text x = 2.\text k.\pi + \frac{\pi}{2} \\\\\ \boxed{\text x = \text k.\pi + \frac{\pi}{4} }$

Substituindo na 1ª equação para achar y :

$\displaystyle \text x+\text y = 2.\text k.\pi \\\\\\ \text k.\pi +\frac{\pi}{4} + \text y = 2\text k.\pi \\\\\\ \text y = 2.\text k.\pi - \text k.\pi - \frac{\pi }{4} \\\\\\ \boxed{\text y =\text k.\pi - \frac{\pi}{4}}$

Soluções :

$\huge\boxed{\ \text x = \text k.\pi + \frac{\pi}{4} \ ; \ \underline{\text{com k} \in \mathbb{Z}}\ }\checkmark \\\\\\ \boxed{\ \text y = \text k.\pi - \frac{\pi}{4} \ ; \ \underline{\text{com k} \in \mathbb{Z}}\ }\checkmark$