Question

Tg x>(√3/3) ctg x<(-1) cos≥(√2/2) sinx≤(-√3/2) Помогите пожалуйста

Answer 1

1)  tgx>√3/3
 π/6+πn<x<π/2+πn,n∈z
2)ctgx<-1
3π/4+πn<x<π+πn,n∈z
3) cosx≥√2/2
  -π/4+2πn≤x≤π/4+2πn,n∈z
4) sinx≤-√3/2
  -2π/3+2πn≤x≤-π/3+2πn,n∈z

Answer 2

Solving the Inequalities

Let's solve the given inequalities step by step:

1. Tg(x) > (√3/3)

To solve this inequality, we need to find the values of x for which the tangent of x is greater than (√3/3).

2. Ctg(x) < (-1)

Similarly, we need to find the values of x for which the cotangent of x is less than -1.

3. Cos(x) ≥ (√2/2)

We need to find the values of x for which the cosine of x is greater than or equal to (√2/2).

4. Sin(x) ≤ (-√3/2)

Lastly, we need to find the values of x for which the sine of x is less than or equal to -√3/2.

Let's solve each inequality one by one.

Solving Tg(x) > (√3/3)

To solve the inequality Tg(x) > (√3/3), we can use the inverse tangent function to find the values of x.

Taking the inverse tangent of both sides, we get:

x > arctan(√3/3)

The inverse tangent of (√3/3) is approximately 30 degrees or π/6 radians.

Therefore, the solution to Tg(x) > (√3/3) is x > π/6.

Solving Ctg(x) < (-1)

To solve the inequality Ctg(x) < (-1), we can use the inverse cotangent function to find the values of x.

Taking the inverse cotangent of both sides, we get:

x < arccot(-1)

The inverse cotangent of (-1) is approximately 135 degrees or 3π/4 radians.

Therefore, the solution to Ctg(x) < (-1) is x < 3π/4.

Solving Cos(x) ≥ (√2/2)

To solve the inequality Cos(x) ≥ (√2/2), we can use the inverse cosine function to find the values of x.

Taking the inverse cosine of both sides, we get:

x ≥ arccos(√2/2)

The inverse cosine of (√2/2) is approximately 45 degrees or π/4 radians.

Therefore, the solution to Cos(x) ≥ (√2/2) is x ≥ π/4.

Solving Sin(x) ≤ (-√3/2)

To solve the inequality Sin(x) ≤ (-√3/2), we can use the inverse sine function to find the values of x.

Taking the inverse sine of both sides, we get:

x ≤ arcsin(-√3/2)

The inverse sine of (-√3/2) is approximately -60 degrees or -π/3 radians.

Therefore, the solution to Sin(x) ≤ (-√3/2) is x ≤ -π/3.

Summary of Solutions

Combining the solutions from each inequality, we have:

x > π/6, x < 3π/4, x ≥ π/4, x ≤ -π/3

To find the common solution to all the inequalities, we need to find the intersection of these solution sets.

Taking the intersection of the solution sets, we find that the common solution is:

x > π/6 and x < 3π/4

Therefore, the values of x that satisfy all the given inequalities are x > π/6 and x < 3π/4.

Please note that the solutions provided are based on the given inequalities and the calculations performed.