Sin 3x - 2·sin x ≤ 0 (sin 3x - sin x) - sin x ≤ 0 2·sin x · cos 2x - sin x ≤ 0 sin x·(2·cos 2x - 1) ≤ 0 sin x·(2·(1 - 2·sin²x) - 1) ≤ 0 sin x·(2 - 4·sin²x - 1) ≤ 0 sin x·(1 - 4·sin²x) ≤ 0 sin x·(1 - 2·sin x)·(1 + 2·sin x) ≤ 0 Замена: sin x = t. t·(1 - 2t)·(1 + 2t) ≤ 0
-1/2 0 1/2 ...> t + - + - t ∈ [-1/2; 0] ∪ [1/2; +∞) Делая обратную замену, учитываем, что |sin x| ≤ 1. x ∈ [-π/6 + 2πn; 2πn] ∪ [π/6 + 2πn; 5π/6 + 2πn] ∪ [π + 2πn; 7π/6 + 2πn], n ∈ Z
(sin 3x - sin x) - sin x ≤ 0
2·sin x · cos 2x - sin x ≤ 0
sin x·(2·cos 2x - 1) ≤ 0
sin x·(2·(1 - 2·sin²x) - 1) ≤ 0
sin x·(2 - 4·sin²x - 1) ≤ 0
sin x·(1 - 4·sin²x) ≤ 0
sin x·(1 - 2·sin x)·(1 + 2·sin x) ≤ 0
Замена: sin x = t.
t·(1 - 2t)·(1 + 2t) ≤ 0
-1/2 0 1/2
...> t
+ - + -
t ∈ [-1/2; 0] ∪ [1/2; +∞)
Делая обратную замену, учитываем, что |sin x| ≤ 1.
x ∈ [-π/6 + 2πn; 2πn] ∪ [π/6 + 2πn; 5π/6 + 2πn] ∪ [π + 2πn; 7π/6 + 2πn], n ∈ Z