Dê o conjunto solução da inequação:
cotg(x) + cossec(x)/sen (x) <igual 1,
para <igual x <igual 2pi e sen diferente de 0.
cotg(x) + cossec(x)/sen (x) <igual 1,
para <igual x <igual 2pi e sen diferente de 0.
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cotg (x) + cossec (x)/ sen (x) ≤ 1 ............[0,2π]
cos (x)/sen(x) + 1/sen(x) * 1/ sen (x) ≤ 1
cos (x)/sen(x) + 1/sen²(x) ≤ 1
cos (x)*sen(x)/sen²(x) + 1/sen²(x) -1 ≤ 0
cos (x)*sen(x)/sen²(x) + 1/sen²(x) -sen²(x)/sen²(x) ≤ 0
[cos (x)*sen(x)+ 1 -sen²(x)]/sen²(x) ≤ 0
[cos (x)*sen(x)+ cos²(x)]/sen²(x) ≤ 0
p=cos (x)*sen(x)+ cos²(x)
p=cos(x)*(sen(x)+cos(x))
cos(x) =0 ==> x=0 , x=π, x=2π são as raízes
p ==>0+++++++(π/2)------(π)---------(3π/2)+++++++++2π
q=sen(x)+ cos(x) =0
raízes ==> x=π-π/4 =3π/2 x =2π-π/4 =7π/4
q ==>0++++++++++(3π/4)---------------------(7π/4)+++++2π
r=sen²(x) sempre será positivo, é elevado ao quadrado
r+++++++++++++++++++++++++++++++++++++++++2π
Estudo de sinais:
p-0++++(π/2)-------------------(π)------(3π/2)+++++++++++++++++2π
q-0++++++++++++(3pi/4)-----------------------(7π/4)+++++++++++2π
r-0+++++++++++++++++++++++++++++++++++++++++++++2π
p*q/r++(π/2)--(3π/4)++(π)+++++++(3π/2)--(7pi/4)+++++++2π
(π/2) < x < 3π/4 U 3π/2 < x < 7π/4 é a resposta