Question

Sin(x) * cos(x) * cos(2x) * cos(8x) - 1/4sin(12x) = 0

Answer 1

Ответ:

4sinx*cosx*cos2x*cos8x=sin12x 2sin2x*cos2x*cos8x=sin(4x+8x) sin4x*cos8x=sin4x*cos8x+cos4x*sin8x cos4x*sin8x=0 a) cos4x=0 4x=π/2+πn x1=π/8+πn/4 б) sin8x=0 8x=πn x2=πn/8

Answer 2

To solve the equation sin(x) * cos(x) * cos(2x) * cos(8x) - 1/4 * sin(12x) = 0, we can follow these steps:

Step 1: Simplify the equation Step 2: Find the solutions within a given range

Step 1: Simplify the equation Using trigonometric identities, we can rewrite the equation as:

sin(x) * cos(x) * cos(2x) * cos(8x) - 1/4 * sin(12x) = 0 (sin(2x)/2) * cos(2x) * cos(8x) * (cos(12x) - 1) = 0 (sin(2x)/2) * cos(2x) * cos(8x) * (-2sin^2(6x)) = 0 -1/2 * sin(2x) * cos(2x) * cos(8x) * sin^2(6x) = 0 -1/2 * (sin(4x)/2) * cos(8x) * sin^2(6x) = 0 (-sin(4x)/4) * (1 - cos(16x)) * sin^2(6x) = 0

Now we have two cases:

Case 1: sin(4x) = 0 This gives us the solutions: x = k * pi/4, where k is an integer.

Case 2: 1 - cos(16x) = 0 Solving for cos(16x) = 1, we get 16x = 2 * pi * n, where n is an integer. Dividing both sides by 16, we have x = pi * n / 8, where n is an integer.

Step 2: Find the solutions within a given range If you have a specific range in mind, you can substitute the values of x within that range into the equation to find the solutions. For example, if you want to find the solutions in the range [0, 2pi], you can substitute each value of x in the equation and check if it satisfies the equation.

Please note that these solutions are general solutions, and if you have a specific range or set of values for x, you can further refine the solutions accordingly.