Question

Cos6x-12cos^3x=4-9cosx

Answer 1

$cos6x-12cos^3x=4-9cosx\\cos6x-12cos^3x+9cosx-4=0\\2cos^23x-3cos3x-5=0\\(cos3x+1)(2cos3x-5)=0\\x=\frac{\pi}{3}+\frac{2\pi k}{3} ,k\in \mathbb{Z}$

Answer 2

To solve the equation cos(6x) - 12cos^3(x) = 4 - 9cos(x), we can simplify it by using trigonometric identities and then apply algebraic techniques to solve for x.

Let's simplify the equation step by step:

1. Start by applying the identity cos(2x) = 2cos^2(x) - 1. cos(6x) can be expressed as cos(2 * 3x), so we have: cos(6x) = 2cos^2(3x) - 1

2. Substitute this expression back into the original equation: 2cos^2(3x) - 1 - 12cos^3(x) = 4 - 9cos(x)

3. Rearrange the equation to get all terms on one side: 2cos^2(3x) - 12cos^3(x) - 9cos(x) + 5 = 0

Now, let's solve the simplified equation using algebraic techniques:

4. Factor out cos(x) from the terms that contain cos(x): cos(x)(2cos^2(3x) - 12cos^2(x) - 9) + 5 = 0

5. Factor out cos(x) from the first term inside the parentheses: cos(x)[2cos^2(3x) - (6cos^2(x) + 6sin^2(x)) - 9] + 5 = 0

6. Simplify the expression inside the brackets: cos(x)(2cos^2(3x) - 6cos^2(x) - 6sin^2(x) - 9) + 5 = 0

7. Apply the identity sin^2(x) = 1 - cos^2(x): cos(x)(2cos^2(3x) - 6cos^2(x) - 6(1 - cos^2(x)) - 9) + 5 = 0

8. Simplify the expression inside the parentheses: cos(x)(2cos^2(3x) - 6cos^2(x) - 6 + 6cos^2(x) - 9) + 5 = 0

9. Combine like terms: cos(x)(2cos^2(3x) + 6cos^2(x) - 15) + 5 = 0

10. Factor out a common factor from the terms inside the parentheses: cos(x)(2cos^2(3x) + 6cos^2(x) - 15) + 5 = 0

11. Now, we have two possibilities for the equation to be true:

    First possibility: cos(x) = 0 This gives us x = (2n + 1) * π/2, where n is an integer.

    Second possibility: 2cos^2(3x) + 6cos^2(x) - 15 = 0

To solve the second possibility, we can simplify the equation further:

12. Divide the equation by 2: cos^2(3x) + 3cos^2(x) - 7.5 = 0

13. Multiply the equation by 2 to eliminate fractions: 2cos^2(3x) + 6cos^2(x) - 15 = 0

14. Notice that this equation is the same as the expression inside the parentheses in step 11.

Therefore, the solutions to the equation cos(6x) - 12cos^3(x)