Question

Resolva a equação trigonometrica:
cos (x+ π/6)=-1

Answer 1

$\cos\!\left(x+\dfrac{\pi}{6} \right )=-1\\\\\\ \cos\!\left(x+\dfrac{\pi}{6} \right )=\cos \pi$


Então,

$x+\dfrac{\pi}{6}=\pm\,\pi+k\cdot 2\pi\\\\\\ x=\pm\,\pi-\dfrac{\pi}{6}+k\cdot 2\pi\\\\\\ \begin{array}{rcl} x=\pi-\dfrac{\pi}{6}+k\cdot 2\pi&~\text{ ou }~&x=-\pi-\dfrac{\pi}{6}+k\cdot 2\pi\\\\ x=\dfrac{6\pi}{6}-\dfrac{\pi}{6}+k\cdot 2\pi&~\text{ ou }~&x=-\,\dfrac{6\pi}{6}-\dfrac{\pi}{6}+k\cdot 2\pi\\\\ x=\dfrac{5\pi}{6}+k\cdot 2\pi&~\text{ ou }~&x=-\,\dfrac{7\pi}{6}+k\cdot 2\pi\\\\ \end{array}$


onde $k$ é inteiro.


Conjunto solução:

$S=\left\{x\in\mathbb{R}:~~x=\dfrac{5\pi}{6}+k\cdot 2\pi~\text{ ou }~x=-\,\dfrac{7\pi}{6}+k\cdot 2\pi,\;\text{ com }k\in\mathbb{Z}\right\}$


Bons estudos! :-)