Equação trigonométrica

por mahriana Dom 10 Fev 2013, 15:09

Resolva a inequação

$\cos{4x}> - \frac{1}{2}$ Supondo $x\,\, \in \,\,\,\left[0, 2 \pi \right]$ :

Spoiler:

por **ramonss** Dom 10 Fev 2013, 15:54

$\\\small{\begin{Bmatrix}2k\pi\leq 4x<\frac{2\pi}{3}+2k\pi\Rightarrow \frac{k\pi}{2}\leq x<\frac{\pi}{6}+\frac{k\pi}{2}\quad (I)\\\\\frac{4\pi}{3}+2k\pi< 4x<2\pi+2k\pi\Rightarrow \frac{\pi}{3}+\frac{k\pi}{2}<x<\frac{\pi}{2}+\frac{k\pi}{2}\quad (II)\end{matrix}}\\\\\\ \small{(I):}\\ k=0\rightarrow 0\leq x<\frac{\pi}{6}\\\\k = 1 \rightarrow \frac{\pi}{2}\leq x <\frac{2\pi}{3}\\\\k=2 \rightarrow \pi \leq x <\frac{7\pi}{6}\\\\k=3\rightarrow \frac{3\pi}{2}\leq x< \frac{5\pi}{3}$

$\\(II):\\\\k=0\rightarrow \frac{\pi}{3}<x\leq\frac{\pi}{2}\\\\k=1\rightarrow \frac{5\pi}{6}<x\leq \pi\\\\k=2\rightarrow \frac{4\pi}{3}<x\leq \frac{3\pi}{2}\\\\k=3\rightarrow \frac{11\pi}{6}<x\leq 2\pi$

Temos que

$\\(I) \cap (II) = \left\{ 0 \leq x <\frac{\pi}{6} \,\, ou\,\,\,\frac{\pi}{3}<x<\frac{2\pi}{3} \,\,\,\,\, ou\,\, \frac{5\pi}{6}<x<\frac{7\pi}{6}\,\, ou \,\, \frac{4\pi}{3}<x<\frac{5\pi}{3}\,\,ou \,\, \frac{11\pi}{6}<x\leq 2\pi \right\}$

por **Leonardo Sueiro** Dom 10 Fev 2013, 15:55

4x = y

cos y > -1/2

2π/3 + 2kπ > y ≥ 2kπ ou 2π + 2kπ ≥ y > 4π/3 + 2kπ

2π/3 + 2kπ > 4x ≥ 2kπ ou 2π + 2kπ≥ 4x > 4π/3 + 2kπ

π/6 + kπ/2 > x ≥ kπ/2 (1) ou π/2 + kπ/2 ≥ x > π/3 + kπ/2 (2)

π/6 + kπ/2 > x ≥ kπ/2

k = 0 -> kπ/2 = 0 => π/6 > x ≥ 0
k = 1 -> kπ/2 = π/2 => 2π/3 > x ≥ π/2
k = 2 -> kπ/2 = π => 7π/6 > x ≥ π
k = 3 -> kπ/2 = 3π/2 => 5π/3 > x ≥ 3π/2
k = 4 -> kπ/2 = 2π => π/6 + 2π > x ≥ 2π

π/2 + kπ/2 > x ≥ π/3 + kπ/2

k = 0 -> kπ/2 = 0 => π/2 ≥ x > π/3
k = 1 -> kπ/2 = π/2 => π ≥ x > 5π/6
k = 2 -> kπ/2 = π => 3π/2 ≥ x > 4π/3
k = 3 -> kπ/2 = 3π/2 => 2π ≥ x > 11π/6
k = 4 -> kπ/2 = 2π => 2,5π ≥ x > 7π/3

Basta juntar as inequações!!!

por Luck Dom 10 Fev 2013, 16:00

cos4x > -1/2
-2pi/3 + 2kpi < 4x < 2pi/3 + 2kpi
-2pi/12 + kpi/2 < x < 2pi/12 + kpi/2
(-2pi + 6kpi)/12 < x < (2pi + 6kpi)/12 , k ∈ Z
k = 0 --> 0 =< x < pi/6
k=1 --> pi/3 < x < 2pi/3
k=2 --> 5pi/6 < x < 7pi/6
k=3 -- > 4pi/3 < x < 5pi/3
k=4 --> 11pi/6 < x =< 2pi

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