Question

6sin(x+п\4)<=3 (неравенство)

Answer 1

Sin(x+π/4)≤1/2
5π/6+2πn≤x+π/4≤11π/6+2πn,n∈z
7π/12+2πn≤x≤19π/12+2πn,n∈z

Answer 2

To solve the inequality 6sin(x + π/4) ≤ 3, we can follow these steps:

Step 1: Subtract 3 from both sides to isolate the sine term: 6sin(x + π/4) - 3 ≤ 0

Step 2: Divide both sides by 6 to obtain the coefficient of sin(x + π/4): sin(x + π/4) - 1/2 ≤ 0

Step 3: Determine the critical points by finding the values of x that make sin(x + π/4) - 1/2 equal to zero. We know that sin(π/6) = 1/2, so:

x + π/4 = π/6 x = π/6 - π/4 x = π/12 - π/4 x = π/12 - 3π/12 x = -2π/12 x = -π/6

Step 4: Analyze the sign of sin(x + π/4) - 1/2 in each interval:

Interval 1: (negative infinity, -π/6) Choose a test point, for example, x = -π/4: sin(-π/4 + π/4) - 1/2 = 0 - 1/2 = -1/2 Since -1/2 is less than or equal to 0, this interval satisfies the inequality.

Interval 2: (-π/6, +∞) Choose a test point, for example, x = 0: sin(0 + π/4) - 1/2 = 1/2 - 1/2 = 0 Since 0 is less than or equal to 0, this interval satisfies the inequality.

Step 5: Combine the intervals that satisfy the inequality: (-∞, -π/6] U [-π/6, +∞)

Therefore, the solution to the inequality 6sin(x + π/4) ≤ 3 is (-∞, -π/6] U [-π/6, +∞).