Question

Resolva a equação sen 2 x - sen x = 0, no domínio R.

Answer 1

$\displaystyle \sf sen(2x)-sen(x)=0 \\\\ 2\cdot sen(x)\cdot cos(x)-sen(x)= 0\\\\ sen(x)\cdot (2cos(x)-1) = 0 \\\\ Da{\'i}} : \\\\ sen(x) = 0 \to x=2\cdot k\cdot \pi \ , \ x=\pi + 2\cdot k\pi \ \ ; \ k\in\mathbb{Z}\\\\\\ 2cos(x)-1=0 \to 2cos(x)=1 \to cos(x) =\frac{1}{2}\\\\\\ cos(x) = \frac{1}{2} \to x=\frac{\pi }{3} + 2\cdot k\cdot \pi \ ,\ x=\frac{5\pi }{3}+2\cdot k\cdot \pi \\\\\ Portanto :$

$\displaystyle \sf x = 2\cdot k\cdot \pi \ ; \ k\in\mathbb{Z} \\\\ x = \pi + 2\cdot k\cdot \pi \ ; \ k\in\mathbb{Z} \\\\ x = \frac{\pi }{3}+2\cdot k\cdot \pi \ ; \ k\in\mathbb{Z} \\\\\\ x = \frac{5\pi }{3}+2\cdot k\cdot \pi \ ; \ k\in\mathbb{Z}$