Question

Resolva a equação |senX| = tg X

Answer 1

Pela igualdade do módulo de um número real a outro número real

$|\sin{x}|=\tan{x}~\Leftrightarrow~\sin{x}=\tan{x}~\lor~\sin{x}=-\tan{x}$

O caso $\sin{x}=\tan{x}$:

$\sin{x}=\tan{x}~\Longrightarrow~\sin{x}=\dfrac{\sin{x}}{\cos{x}}~\Longrightarrow~\sin{x}\cos{x}=\sin{x}\\\\\Longrightarrow~\sin{x}\cos{x}-\sin{x}=0~\Longrightarrow~\sin{x}(\cos{x}-1)=0\\\\\Longrightarrow~\sin{x}=0~\lor~\cos{x}=1\\\\S_{1}=\{x\in\mathbb{R}:x=k\pi\},~k\in\mathbb{Z}$

O caso $\sin{x}=-\tan{x}$:

$\sin{x}=-\tan{x}~\Longrightarrow~\sin{x}=-\dfrac{\sin{x}}{\cos{x}}~\Longrightarrow~\sin{x}\cos{x}=-\sin{x}\\\\\Longrightarrow~\sin{x}\cos{x}+\sin{x}=0~\Longrightarrow~\sin{x}(\cos{x}+1)=0\\\\\Longrightarrow~\sin{x}=0~\lor~\cos{x}=-1\\\\S_{2}=\{x\in\mathbb{R}:x=k\pi\},~k\in\mathbb{Z}$


$S=S_{1} \cup S_{2}=\{x\in\mathbb{R}:x=k\pi\},~k\in\mathbb{Z}$