Question

O conjunto solução da equação cos x = cos (π/3 – x ), para 0<x<2π é correspondente a qual valor?​

Answer 1

$\mathsf{cos(\alpha)=cos(\beta)}\huge\Leftrightarrow\begin{cases}\mathsf{\alpha=\beta}\\\mathsf{\alpha=-\beta}\end{cases}$

$\mathsf{cos(x)=cos(\dfrac{\pi}{3}-x)}\Leftrightarrow\begin{cases}\mathsf{x=\dfrac{\pi}{3}-x}\\\mathsf{x=-(\dfrac{\pi}{3}-x)}\end{cases}$

$\mathsf{x=\dfrac{\pi}{3}-x\times(3)}\\\mathsf{3x=\pi-3x}\\\mathsf{3x+3x=\pi}\\\mathsf{6x=pi}\\\huge\boxed{\boxed{\mathsf{x=\dfrac{\pi}{6}}}}$

$\mathsf{x=-(\dfrac{\pi}{3}-x)}\\\mathsf{x=-\dfrac{\pi}{3}+x\to\not\exists~x\in\mathbb{R}}$

$\huge\boxed{\boxed{\mathsf{s=\{\dfrac{\pi}{6}\}}}}$