Question

O conjunto solução da equação cos x = cos (π/3 – x ), para 0

Answer 1

$\mathsf{cos(a)=cos(b)}\to\begin{cases}\mathsf{a=b}\\\mathsf{a=-b}\end{cases}$

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

$\begin{cases}\mathsf{x=\dfrac{\pi}{3}-x}\\\mathsf{x=-\dfrac{\pi}{3}+x}\end{cases}$

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

$\mathsf{6x=\pi}\\\mathsf{x=\dfrac{\pi}{6}}$

Existem outras soluções que são arcos relacionados a $\mathsf{\dfrac{\pi}{6}}$

Os arcos são $\mathsf{\dfrac{5\pi}{6}}$ $\mathsf{\dfrac{7\pi}{6}}$

e $\mathsf{\dfrac{11\pi}{6}}$

Portanto

$\huge\boxed{\boxed{\mathsf{s=\{\dfrac{\pi}{6},\dfrac{5\pi}{6},\dfrac{7\pi}{6},\dfrac{11\pi}{6}\}}}}$