Question

Sin^2x>=1/4 розв'язати ​

Answer 1

Ответ:

Объяснение:

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Answer 2

To solve the inequality sin^2(x) >= 1/4, we need to find the range of values for x that satisfy the inequality.

Let's break it down step by step:

1. Rewrite the inequality using the trigonometric identity sin^2(x) = (1 - cos^2(x)):

(1 - cos^2(x)) >= 1/4

2. Now, bring all terms to one side of the inequality to set it to zero:

1 - cos^2(x) - 1/4 >= 0

3. Combine the constants:

3/4 - cos^2(x) >= 0

4. Rearrange the terms:

cos^2(x) - 3/4 <= 0

5. Factorize the left side:

(cos(x) - √3/2)(cos(x) + √3/2) <= 0

6. Now, consider the sign of each factor to determine the solution:

For the inequality to be less than or equal to zero, one of the factors must be negative or zero while the other is positive or zero.

a) cos(x) - √3/2 <= 0 b) cos(x) + √3/2 >= 0

7. Solve each factor separately:

a) cos(x) - √3/2 <= 0: cos(x) <= √3/2

b) cos(x) + √3/2 >= 0: cos(x) >= -√3/2

8. The solutions for x in the range 0 <= x <= 2π (one full period of the sine function) that satisfy the inequality are obtained from the intersection of the solutions for a) and b) above:

0 <= x <= π/6 or 5π/6 <= x <= 2π

So, the solution to the inequality sin^2(x) >= 1/4 is x belonging to the intervals [0, π/6] and [5π/6, 2π].