Question

Qual a solucao em IR (reais) de sen x = -1/2

Answer 1

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$\boxed{\begin{array}{c}\sf a~func_{\!\!,}\tilde ao~seno~\acute e~negativa~no~3^{\underline o}~e~4^{\underline o}~quadrantes\\\sf o~arco~cujo~seno~vale~\dfrac{1}{2}~\acute e~\dfrac{\pi}{6}.\\\sf~vamos~descobrir~os~arcos~c\hat ongruos~a~\dfrac{\pi}{6}~no~3^{\underline o}~e~4^{\underline o}~quadrantes.\end{array}}\\\underline{\sf no~3^{\underline o}~quadrante:}\\\sf \pi+\dfrac{\pi}{6}=\dfrac{6\pi+\pi}{6}=\dfrac{7\pi}{6}\\\underline{\sf no~4^{\underline o}~quadrante:}$

$\sf2\pi-\dfrac{\pi}{6}=\dfrac{12\pi-\pi}{6}=\dfrac{11\pi}{6}$

$\underline{\sf DA\acute I:}\\\sf sen(x)=-\dfrac{1}{2}\\\sf sen(x)=sen\left(\dfrac{7\pi}{6}\right)\\\sf x=\dfrac{7\pi}{6}+2k\pi~k\in\mathbb{Z}\\\sf sen(x)=-\dfrac{1}{2}\\\sf sen(x)=sen\left(\dfrac{11\pi}{6}\right)\\\sf x=\dfrac{11\pi}{6}+2k\pi~k\in\mathbb{Z}\\\boxed{\boxed{\boxed{\boxed{\sf S=\left\{x\in\mathbb{R}/ x=\dfrac{7\pi}{6}+2k\pi~k\in\mathbb{Z},x=\dfrac{11\pi}{6}+2k\pi,k\in\mathbb{Z}\right\}}}}}$