Question

Dê o conjunto solução da inequação:
cotg(x) + cossec(x)/sen (x) <igual 1,
 para <igual x <igual 2pi e sen diferente de 0.

Answer 1

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