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 3
Ответ: [-2π-6 + 4πn; 2π-6 + 4πn], n∈Z
3 3
б) sin (x/4 -2) < <u>√2
2
- 5π + 2πn < x/4 -2 < π/4 + 2πn, n∈Z<br> 4
- 5π +2 +2πn < x/4 <<u> π + 2 + 2πn
4 4
-5π+8 + 2πn < x/4 < <u>π+8 +2πn
4 4
-5π +8 + 8πn < x < π+8 +8πn, n∈Z<br>Ответ: (-5π+8+8πn; π+8+8πn), n∈Z
в) 2cos² x +sinx - 1 < 0
2(1-sin² x) + sinx -1 <0<br>2 - 2sin² x +sinx -1 < 0
-2sin² x + sinx + 1 <0<br>2sin² x - sinx -1 > 0
Замена y=sinx
2y² -y-1>0
2y² -y -1=0
D=1+8=9
y₁ = 1-3 = -2/4 = -1/2
4
y₂ = 1+3 = 1
4
+ - +
----------- -1/2 ------------ 1 -------------
\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\
{y< -1/2
{y> 1
{sinx < -1/2
{sinx > 1
sinx < -1/2
-5π + 2πn < x <<u> -π + 2πn, n∈Z
6 6
sinx > 1
нет решений
Ответ: (-5π + 2πn; -π +2πn), n∈Z
6 6
г) 2sin² x - 5cosx +1 >0
2(1-cos² x) -5cosx +1 >0
2 - 2cos² x -5cosx +1 >0
-2cos² x - 5cosx + 3 >0
2cos² x + 5cosx -3 < 0
Замена y=cosx
2y² + 5y -3 < 0
2y² +5y -3 =0
D=25 + 24=49
y₁ = -5-7 = -3
4
y₂ = -5+7 = 2/4 = 1/2
4
+ - +
---------- -3 ------------ 1/2 ----------
\\\\\\\\\\\\\\\
{y > -3
{y < 1/2
{cosx > -3
{cosx < 1/2
cosx > -3
x∈R - любое действительное число
cosx < 1/2
π + 2πn < x < 5π + 2πn, n∈Z
3 3
Ответ: (π + 2πn; 5π + 2πn), n∈Z
3 3