General Solution for cos(x)cos(2x)cos(3x) = 1/4:

To find the general solution for the equation cos(x)cos(2x)cos(3x) = 1/4, we can follow the steps below:

Step 1: Simplify the given equation:

Using the trigonometric identity cos(2θ) = 2cos^2(θ) - 1, we can rewrite the equation as:

cos(x)(2cos^2(2x) - 1)cos(3x) = 1/4

Simplifying further, we have:

2cos(x)cos^2(2x)cos(3x) - cos(x)cos(3x) = 1/4

Step 2: Apply the double-angle formula:

The double-angle formula for cosine states that cos(2θ) = 2cos^2(θ) - 1. We can use this formula to simplify the equation even further:

2cos(x)[2cos^2(x) - 1]^2cos(3x) - cos(x)cos(3x) = 1/4

Expanding the square term, we get:

2cos(x)[4cos^4(x) - 4cos^2(x) + 1]cos(3x) - cos(x)cos(3x) = 1/4

Simplifying, we have:

8cos^5(x)cos(3x) - 8cos^3(x)cos(3x) + 2cos(x)cos(3x) - cos(x)cos(3x) = 1/4

Step 3: Combine like terms:

Combining the like terms, we get:

8cos^5(x)cos(3x) - 8cos^3(x)cos(3x) + cos(x)cos(3x) = 1/4

Step 4: Factor out common terms:

Factoring out cos(3x), we get:

cos(3x)[8cos^5(x) - 8cos^3(x) + cos(x)] = 1/4

Step 5: Solve for cos(3x):

To find the values of cos(3x) that satisfy the equation, we set the equation inside the brackets equal to 1/4:

8cos^5(x) - 8cos^3(x) + cos(x) = 1/4

Step 6: Solve the polynomial equation:

Solving the polynomial equation 8cos^5(x) - 8cos^3(x) + cos(x) = 1/4 can be challenging due to its complexity. The general solution for this equation involves using numerical methods such as graphing or iterative approximation techniques to find the approximate values of x that satisfy the equation.

Step 7: Substitute x to find cos(3x):

Once we find the values of x that satisfy the polynomial equation, we can substitute those values back into the equation cos(3x)[8cos^5(x) - 8cos^3(x) + cos(x)] = 1/4 to find the corresponding values