2 cos^2=1 [0, 2pi)
To solve the equation 2cos^2(x) = 1 in the interval [0, 2π), you can follow these steps:
Step 1: Rewrite the equation using the double-angle identity for cosine:
cos(2x) = 1/2
Step 2: Recall that the solutions of a cosine function occur when the angle corresponds to the given output. So, cos(2x) = 1/2 means that the angle (2x) has a cosine value of 1/2.
Step 3: Find the reference angle. Since cos(π/3) = 1/2, we can use this as our reference angle.
Step 4: Determine the solutions within the given interval [0, 2π) by considering the possible values of (2x), based on the reference angle.
- First solution: 2x = π/3 => x = π/6
- Second solution: 2x = 5π/3 => x = 5π/6
Therefore, the solutions to the equation 2cos^2(x) = 1 in the interval [0, 2π) are x = π/6 and x = 5π/6.