а) cos(x/2 +1) ≥ 1/2- π/3 + 2πn ≤ x/2 +1 ≤ π/3 + 2πn, n∈Z-π/3 - 1 +2πn ≤ x/2 ≤ π/3 - 1 +2πn-
π-3 +2πn ≤ x/2 ≤
π-3 + 2πn 3 3-
2π-6 +4πn ≤ x ≤
2π-6 + 4πn, n∈Z 3 3Ответ: [
-2π-6 + 4πn;
2π-6 + 4πn], n∈Z 3 3б) sin (x/4 -2) <
√2 2-
5π + 2πn < x/4 -2 < π/4 + 2πn, n∈Z 4
- 5π +2 +2πn < x/4 <
π + 2 + 2πn 4 4-
5π+8 + 2πn < x/4 <
π+8 +2πn 4 4-5π +8 + 8πn < x < π+8 +8πn, n∈ZОтвет: (-5π+8+8πn; π+8+8πn), n∈Zв) 2cos² x +sinx - 1 < 02(1-sin² x) + sinx -1 <02 - 2sin² x +sinx -1 < 0-2sin² x + sinx + 1 <02sin² x - sinx -1 > 0Замена y=sinx2y² -y-1>02y² -y -1=0D=1+8=9y₁ =
1-3 = -2/4 = -1/2 4y₂ =
1+3 = 1 4 + - +----------- -1/2 ------------ 1 -------------\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\{y< -1/2{y> 1{sinx < -1/2{sinx > 1sinx < -1/2
-5π + 2πn < x <
-π + 2πn, n∈Z 6 6sinx > 1нет решенийОтвет: (
-5π + 2πn;
-π +2πn), n∈Z 6 6г) 2sin² x - 5cosx +1 >02(1-cos² x) -5cosx +1 >02 - 2cos² x -5cosx +1 >0-2cos² x - 5cosx + 3 >02cos² x + 5cosx -3 < 0Замена y=cosx2y² + 5y -3 < 02y² +5y -3 =0D=25 + 24=49y₁ =
-5-7 = -3 4y₂ =
-5+7 = 2/4 = 1/2 4 + - +---------- -3 ------------ 1/2 ---------- \\\\\\\\\\\\\\\{y > -3{y < 1/2{cosx > -3{cosx < 1/2cosx > -3x∈R - любое действительное числоcosx < 1/2
π + 2πn < x <
5π + 2πn, n∈Z 3 3Ответ: (
π + 2πn;
5π + 2πn), n∈Z 3 3