Question

A soma das raízes da equação trigonométrica da imagem anexada é:

Answer 1

$\mathrm{sen^3\,}x-\mathrm{sen\,}x=0\\\\ -\mathrm{sen\,}x+\mathrm{sen^3\,}x=0\\\\ -\mathrm{sen\,}x\cdot (1-\mathrm{sen^2\,}x)=0\\\\ \mathrm{sen\,}x\cdot (1-\mathrm{sen^2\,}x)=0\\\\ \mathrm{sen\,}x\cdot \cos^2 x=0\\\\ \begin{array}{rcl} \mathrm{sen\,}x=0&~\text{ ou }~&\cos^2 x=0\\\\ \mathrm{sen\,}x=0&~\text{ ou }~&\cos x=0\\\\ x\in \{0,\pi,\,2\pi\}&~\text{ ou }~&x \in\left\{\dfrac{\pi}{2},\,\dfrac{3\pi}{2} \right \} \end{array}\\\\\\ \therefore~~x\in\left\{0,\,\dfrac{\pi}{2},\,\pi,\,\dfrac{3\pi}{2},\,2\pi \right \}$


A soma das soluções no intervalo de interesse é

$0+\dfrac{\pi}{2}+\pi+\dfrac{3\pi}{2}+2\pi\\\\\\ =\dfrac{\pi+2\pi+3\pi+4\pi}{2}\\\\\\ =\dfrac{10\pi}{2}\\\\\\ =5\pi$


Nenhuma das alternativas apresentadas está correta.


Bons estudos! :-)