Question

Resolva a equação no intervalo $0 \leq x \leq 2\pi$

$2sen( x + \frac{\pi }{2} ) =1$

Answer 1

$\displaystyle 2.\text{sen}(\text x+\frac{\pi}{2}) = 1 \ , \ \text{intervalo : } 0\leq \text x\leq 2\pi$

Temos :

$\displaystyle 2.\text{sen}(\text x+\frac{\pi}{2}) = 1 \\\\\\ \text{sen}(\text x+\frac{\pi}{2}) = \frac{1}{2}$

1º quadrante :

$\displaystyle \text{sen}(\text x+\frac{\pi}{2}) = \text{sen}(\frac{\pi}{6})$

$\displaystyle \text x +\frac{\pi}{2} = \frac{\pi}{6}$

$\displaystyle \text x = \frac{\pi}{6} - \frac{\pi}{2} \\\\\\ \text x = \frac{\pi}{6}-\frac{3\pi }{6} \\\\\\ \text x = \frac{-2\pi}{6} \\\\\\ \text x = \frac{-\pi}{3}}$

sinal negativo significa que é o arco no sentido horário do circulo trigonométrico, sabendo disso, temos :

$\displaystyle \text x = \frac{-\pi}{3}} \\\\\\ \text x = 2\pi -\frac{\pi}{3}} \\\\\ \boxed{\text x = \frac{5\pi }{3}}$

2º quadrante :

$\displaystyle \text{sen}(\text x+\frac{\pi}{2}) = \text{sen}(\pi -\frac{\pi}{6}) \\\\\\ \text{sen}(\text x+\frac{\pi}{2}) = \text{sen}(\frac{5\pi}{6}) \\\\\\ \text x + \frac{\pi}{2} = \frac{5\pi }{6} \\\\\\ \text x = \frac{5\pi}{6} - \frac{\pi}{2} \\\\\\ \text x = \frac{5\pi}{6}-\frac{3\pi }{6} \\\\\\ \boxed{\text x = \frac{\pi}{3}}$

Soluções :

$\huge\boxed{\ \text x=\frac{\pi}{3} \ \ \ \ \text{ou} \ \ \ \text x = \frac{5\pi}{3}\ }} \checkmark$