Question

1) sinx < 1/2 2) cos2x > 0 3) 3 – 4cos2x > 0 4) cosx > – 1/2 5) sin3x < 0 6) tgх> 7) sinx < cosx 8) 1 – 4sin2 x < 0

Answer 1

1. sinx = 1/2,

x = (-1)ⁿ · π/6 + πn, n ∈ Z

Ответ: (-1)ⁿ · π/6 + πn, n ∈ Z.

Answer 2

Let's solve each of these inequalities step by step:

1. sin(x) < 1/2: To solve this inequality, we need to find the values of x for which sin(x) is less than 1/2.

sin(x) < 1/2 when x is in the interval (-π/6, π/6) or (5π/6, 7π/6).

2. cos(2x) > 0: To solve this inequality, we need to find the values of x for which cos(2x) is greater than 0.

cos(2x) > 0 when x is in the intervals (-π/4 + πn, π/4 + πn), where n is an integer.

3. 3 - 4cos(2x) > 0: To solve this inequality, we need to find the values of x for which 3 - 4cos(2x) is greater than 0.

3 - 4cos(2x) > 0 when cos(2x) < 3/4. Using the previous result, we can find the intervals for x.

x is in the intervals (-π/3 + πn, π/3 + πn), where n is an integer.

4. cos(x) > -1/2: To solve this inequality, we need to find the values of x for which cos(x) is greater than -1/2.

cos(x) > -1/2 when x is in the intervals (-2π/3 + 2πn, 2π/3 + 2πn), where n is an integer.

5. sin(3x) < 0: To solve this inequality, we need to find the values of x for which sin(3x) is less than 0.

sin(3x) < 0 when x is in the intervals (2πn, π/3 + 2πn) or (2π/3 + 2πn, π + 2πn), where n is an integer.

6. tan(x) > 0: To solve this inequality, we need to find the values of x for which tan(x) is greater than 0.

tan(x) > 0 when x is in the intervals (nπ, π/2 + nπ), where n is an odd integer.

7. sin(x) < cos(x): To solve this inequality, we need to find the values of x for which sin(x) is less than cos(x).

sin(x) < cos(x) when x is in the intervals (π/4 + πn, 3π/4 + πn), where n is an integer.

8. 1 - 4sin^2(x) < 0: To solve this inequality, we need to find the values of x for which 1 - 4sin^2(x) is less than 0.

1 - 4sin^2(x) < 0 when sin^2(x) > 1/4, which is true for all values of x.

So, the solution to this inequality is all real values of x.

Please note that the intervals are given in terms of n, which represents an integer. The intervals repeat periodically as n varies.