Question

√3 sin2x+cos2x<=-√3 sin^2x-sinxcosx-2cos^2x>0 lctgxl<1

Answer 1

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Answer 2

To analyze the inequality √3 sin(2x) + cos(2x) ≤ -√3 sin^2(x) - sin(x)cos(x) - 2cos^2(x) > 0 and the additional condition lctg(x) < 1, let's break it down step by step.

Step 1: Simplify the inequality and the condition. √3 sin(2x) + cos(2x) ≤ -√3 sin^2(x) - sin(x)cos(x) - 2cos^2(x) > 0 lctg(x) < 1

Step 2: Use trigonometric identities to simplify the expressions. √3 sin(2x) = 2√3 sin(x) cos(x) cos(2x) = cos^2(x) - sin^2(x) = cos^2(x) - (1 - cos^2(x)) = 2cos^2(x) - 1

Substituting these identities into the inequality: 2√3 sin(x) cos(x) + (2cos^2(x) - 1) ≤ -√3 sin^2(x) - sin(x)cos(x) - 2cos^2(x) > 0

Simplifying further: 2√3 sin(x) cos(x) + 2cos^2(x) - 1 ≤ -√3 sin^2(x) - sin(x)cos(x) - 2cos^2(x) > 0

Step 3: Combine like terms: 3√3 sin^2(x) - 3sin(x)cos(x) - 4cos^2(x) + 1 > 0

Step 4: Rearrange the inequality: 3√3 sin^2(x) - 3sin(x)cos(x) - 4cos^2(x) + 1 - 0

Step 5: Simplify the condition: lctg(x) < 1

Step 6: Analyze the solutions: To find the solutions to this inequality and the condition, we need to solve the equations individually and then find the overlapping solution set.

For lctg(x) < 1: This condition states that the cotangent of x must be less than 1. The cotangent function is the reciprocal of the tangent function, so lctg(x) = cos(x) / sin(x). When the cotangent is less than 1, it means that cos(x) is less than sin(x).

For √3 sin^2(x) - 3sin(x)cos(x) - 4cos^2(x) + 1 > 0: We can rewrite this as a quadratic equation: (√3 sin(x) - 4cos(x))(sin(x) + cos(x) - 1) > 0

From this quadratic equation, we get two factors: (√3 sin(x) - 4cos(x)) and (sin(x) + cos(x) - 1).

Now, we need to analyze the signs of these factors to determine the solution.

• When (√3 sin(x) - 4cos(x)) > 0 and (sin(x) + cos(x) - 1) > 0, the quadratic equation is greater than 0.
• When (√3 sin(x) - 4cos(x)) < 0 and (sin(x) + cos(x) - 1) < 0, the quadratic equation is greater than 0.

Step 7: Analyzing the signs of the factors: To analyze the signs, we can use the unit circle or trig