Question

O módulo da soma dos pontos de intersecção das curvas y = sen(x) e z = cos(x), em 0 ≤ x ≤ 2p, é:

Answer 1

$\mathrm{y=\sin{x}\ \ \| \ \ z=\cos{x}\ \ \| \ \ 0\leq x\leq2\pi}\\\\ \mathrm{y=z \ \to\ \sin{x}=\cos{x}\ \to\ \dfrac{\sin{x}}{\cos{x}}=\dfrac{\cos{x}}{\cos{x}}}\\\\ \mathrm{\tan{x}=1\ \to\ x=\arctan{1}\ \to\ x=\dfrac{\pi}{4}+\pi k}\\\\ \mathbf{Para\ 0\leq x\leq2\pi:}\\\\ \mathrm{x_1=\dfrac{\pi}{4}+\pi.0=\dfrac{\pi}{4}\ \ \bigg\| \ \ x_2=\dfrac{\pi}{4}+\pi.1=\dfrac{5\pi}{4}}\\\\ \textbf{Soma das solu\c{c}\~oes:}\\\\ \mathrm{S=\dfrac{\pi}{4}+\dfrac{5\pi}{4}=\dfrac{6\pi}{4}\ \to\ \mathbf{S=\dfrac{3\pi}{2}}}$