Question

Sin3xcosx-cos3xsinx≤0,5

Answer 1

Пошаговое объяснение:

Ответ и решение в файле. Честно говоря, не уверенна в решении

Answer 2

To solve the inequality sin(3x)cos(x) - cos(3x)sin(x) ≤ 0.5, we can simplify the left-hand side and then find the interval(s) where the inequality holds true.

Step 1: Use trigonometric identities to simplify the left-hand side: sin(3x)cos(x) - cos(3x)sin(x) = sin(3x - x) = sin(2x).

So, the inequality becomes sin(2x) ≤ 0.5.

Step 2: Find the interval(s) where the inequality holds true: To solve sin(2x) ≤ 0.5, we'll consider the unit circle and the values of sin(2x) in each quadrant.

In the first and fourth quadrants (0 ≤ 2x ≤ π), sin(2x) is positive, and it reaches a maximum of 1 at π/2. Therefore, sin(2x) ≤ 0.5 in the first and fourth quadrants when 0 ≤ 2x ≤ π/6.

In the second and third quadrants (π ≤ 2x ≤ 2π), sin(2x) is negative, and it reaches a minimum of -1 at 3π/2. Therefore, sin(2x) ≤ 0.5 in the second and third quadrants when 11π/6 ≤ 2x ≤ 2π.

So, the solution to the inequality is: 0 ≤ 2x ≤ π/6 (for the first and fourth quadrants) or 11π/6 ≤ 2x ≤ 2π (for the second and third quadrants).

Finally, divide the interval by 2 to find the solution for x: 0 ≤ x ≤ π/12 (for the first and fourth quadrants) or 11π/12 ≤ x ≤ π (for the second and third quadrants).

Therefore, the solution to the inequality sin(3x)cos(x) - cos(3x)sin(x) ≤ 0.5 is: 0 ≤ x ≤ π/12 or 11π/12 ≤ x ≤ π.