How do I find the solutions for cos2x - cos6x = 0?
To find the solutions for the equation cos2x - cos6x = 0, we need to determine the values of x that satisfy this equation. Let's break it down step by step:
Step 1: Use the cosine double-angle formula
The double-angle formula for cosine states that cos(2θ) = 2cos^2(θ) - 1. We can apply this formula to our equation by replacing cos2x with its equivalent expression:
2cos^2(x) - 1 - cos6x = 0.
Step 2: Simplify the equation
Now, rearrange the terms to obtain a quadratic equation:
2cos^2(x) - cos6x - 1 = 0.
Step 3: Factor the quadratic equation
To solve this quadratic equation, it would be helpful to factor it. However, factoring may not always be straightforward. In such cases, you can use the quadratic formula. The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a).
In our quadratic equation 2cos^2(x) - cos6x - 1 = 0, the coefficients are a = 2, b = -1, and c = -1. Therefore, applying the quadratic formula, we have:
cos(x) = [-(−1) ± √((−1)^2 - 4(2)(−1))] / (2(2)).
cos(x) = [1 ± √(1 + 8)] / 4.
Step 4: Solve for x
To solve for x, we need to consider both the plus and minus signs when applying the quadratic formula. We have:
cos(x) = (1 + √9) / 4 = (1 + 3) / 4 = 4/4 = 1.
cos(x) = (1 - √9) / 4 = (1 - 3) / 4 = -2/4 = -1/2.
Step 5: Determine the angle values
The next step is to find the corresponding angle values for the cosine values obtained in step 4. To do this, we can use the unit circle or a calculator.
For cos(x) = 1, the angle x is 0 degrees or 2π radians.
For cos(x) = -1/2, the angle x is approximately 120 degrees or 2π/3 radians and 240 degrees or 4π/3 radians.
Hence, the solutions for the equation cos2x - cos6x = 0 are x = 0, x = 2π/3, and x = 4π/3.
Remember to always verify the solutions by substituting them back into the original equation to confirm that they satisfy the equation.