Question

Решите неравенство:1) sin 5x cos 3x - cos 5x sin 3x > 1/22) cos x cos x/2 + sin x sin x/2 < -2/7Помогите пожалуйста:)

Answer 1

1) sin5x cos3x - cos5x sin 3x > 1/2

sin(5x-3x) > 1/2

sin 2x > 1/2

 п/6 + 2пN < 2x < п - П/6 + 2пN , N принадлежит Z

 п/6 + 2пN < 2x < 5 П/6 + 2пN N принадлежит Z

п/12 + 2пN < x < 5 П/12 + 2пN N принадлежит Z

Answer 2

Solving the inequality: sin(5x) + cos(3x) - cos(5x) + sin(3x) > 1/2

To solve the inequality sin(5x) + cos(3x) - cos(5x) + sin(3x) > 1/2, we can follow these steps:

Step 1: Combine like terms sin(5x) + sin(3x) + cos(3x) - cos(5x) > 1/2

Step 2: Rearrange the terms sin(5x) - cos(5x) + sin(3x) + cos(3x) > 1/2

Step 3: Use trigonometric identities to simplify the expression 2sin((5x + 3x)/2)cos((5x - 3x)/2) + 2cos((5x + 3x)/2)sin((5x - 3x)/2) > 1/2

Step 4: Simplify further 2sin(4x)cos(x) + 2cos(4x)sin(x) > 1/2

Step 5: Combine like terms 2sin(4x)cos(x) + 2sin(x)cos(4x) > 1/2

Step 6: Use the double-angle formula for sine sin(2x) = 2sin(x)cos(x)

2sin(2x)cos(2x) > 1/2

Step 7: Simplify further sin(2x)cos(2x) > 1/4

Step 8: Use the double-angle formula for cosine cos(2x) = 1 - 2sin^2(x)

sin(2x)(1 - 2sin^2(x)) > 1/4

Step 9: Distribute and rearrange the terms sin(2x) - 2sin^3(2x) > 1/4

Step 10: Simplify further sin(2x) - 2sin^3(2x) - 1/4 > 0

Now, we have a cubic inequality that we need to solve. However, finding the exact solutions for this inequality can be quite complex. It is recommended to use numerical methods or graphing software to approximate the solutions.

Please note that without additional constraints or specific intervals for the variable x, it is not possible to find a precise solution. If you have any additional information or constraints, please provide them, and I will be happy to assist you further.