Question

Determine o conjunto solucão da equação 2 sen² x + 3 sen x - 2 = 0

Answer 1

$\displaystyle \sf 2\cdot sen^2x+3\cdot sen\ x-2 = 0 \\\\ sen\ x = \frac{-3\pm\sqrt{3^2-4\cdot2\cdot(2)}}{2\cdot 2} \\\\\\ sen\ x = \frac{-3\pm\sqrt{9+16}}{4}\\\\\\ sen x = \frac{-3\pm5}{4}\\\\\\ sen x =\frac{-3-5}{4}=-2 \ (\text{N{\~A}O CONV{\'E}M)} \\\\\\ sen\ x =\frac{-3+5}{4}=\frac{1}{2} \\\\\\ sen\ x = \frac{1}{2} \\\\\\ sen \ x = sen\ \left(\frac{\pi}{6}+2\cdot k\cdot \pi \right)\\\\\\ ou \\\\\\ sen\ x = sen\ \left(\frac{5\pi}{6}+2\cdot k\cdot \pi \right)$

Portanto o conjunto solução é :

$\boxed{\displaystyle \sf x =\left(\frac{\pi}{6}+2\cdot k\cdot \pi\right) \ , \ k \in\mathbb{Z}\ ; \ \ x =\left(\frac{5\pi}{6}+2\cdot k\cdot \pi\right) \ , \ k \in\mathbb{Z} } }\checkmark$