Question

Considere a equação:
Sen³x • cosx – senxcos³x = ¼
A soma das raízes da equação para x€[0,2π] é:

(A) 9π/2. (B)4π. (C)3π. (D)2π. (E)5π/2.

Answer 1

$sen^{3}x.cosx - senxcos^{3}x = \frac{1}{4}$

$senxcosx(sen^{2}x - cos^{2}x) = \frac{1}{4}$

$senxcosx(-sen^{2}x + cos^{2}x) = \frac{-1}{4}$

$senxcosx(cos^{2}x - sen^{2}x) = \frac{-1}{4}$

$senxcosx(cos2x) = \frac{-1}{4}$

$2senxcosx(cos2x) = \frac{-1}{2}$

$(sen2x)(cos2x) = \frac{-1}{2}$

$2sen2xcos2x = -1$

$sen4x = -1$

$sen4x = sen\frac{3\pi}{2}$

$4x = \frac{3\pi}{2} + 2k\pi$

$x = \frac{3\pi}{8} + \frac{k\pi}{2}$

$Para k=0 -> x=\frac{3\pi}{8}$

$Para k=1 -> x=\frac{7\pi}{8}$

$Para k=2 -> x=\frac{11\pi}{8}$

$Para k=3 -> x=\frac{15\pi}{8}$

$\frac{3\pi}{8} + \frac{7\pi}{8} + \frac{11\pi}{8} + \frac{15\pi}{8} = \frac{36\pi}{8} = \frac{9\pi}{2}$

$\boxed{Alternativa A}$