Решение:
1) 2 cos x-1 ≥ 0
cosx ≥ 1/2
- arccos (1/2) + 2πn ≤ x ≤ arccos (1/2) + 2πn, n ∈ Z
- π/3 + 2πn ≤ x ≤ π/3 + 2πn, n ∈ Z2) 2sinx + √2 ≥ 0
sinx ≥ - √2/2
arcsin(√2/2) + 2πn ≤ x ≤ π - arcsin(√2/2) + 2πn, n ∈ Z
π/4 + 2πn ≤ x ≤ π - π/4 + 2πn, n ∈ Z
π/4 + 2πn ≤ x ≤ 3π/4 + 2πn, n ∈ Z
3) 2cosx - √3 ≤ 0
2cosx ≤ √3
cosx ≤ √3/2
π/6 + 2πn ≤ x ≤ 2π - π/6 + 2πn, n ∈ Z
π/6 + 2πn ≤ x ≤ 11π/6 + 2πn, n ∈ Z
4) 3tgx + √3 > 0
tgx > - √3/3
arctg(- √3/3) + πn ≤ x ≤ π/2 + πn, n ∈ Z
- π/6 + πn ≤ x ≤ π/2 + πn, n ∈ Z