Solve the equation on the interval [0,2pi). Cos2x=square root 2/2
To solve the equation cos(2x) = sqrt(2)/2 on the interval [0, 2π), we can first find the values of x that satisfy the equation cos(2x) = sqrt(2)/2.
Since cos(2x) = sqrt(2)/2, we can find the first value of x by using the inverse cosine function:
2x = π/4
x = π/8
Now, we know that cosine function has a period of 2π. This means that any value x that satisfies the equation cos(2x) = sqrt(2)/2 can be written as x = (π/8) + (π/2)n, where n is any integer.
In the interval [0, 2π), we need to find all the possible values of x that satisfy the equation:
x = π/8 + (π/2)n
Substitute n = 0, 1, 2, 3 into the equation to get:
x = π/8, 9π/8, 17π/8, 25π/8
Therefore, the solutions in the interval [0, 2π) are:
x = π/8, 9π/8, 17π/8, 25π/8.