Solve the equation on the interval [0,2pi).
cos2x=(*2/2)
To solve the equation cos(2x) = (√2) / 2 on the interval [0, 2π), we can follow these steps:
Step 1: Find the reference angle
Since the cosine function is positive in the first and fourth quadrants, we need to find the reference angle whose cosine equals (√2) / 2. The reference angle with a cosine value of (√2) / 2 is π / 4 (45 degrees).
Step 2: Write down the solutions
Since the cosine function has a period of 2π, we can find additional solutions by adding integer multiples of the period to the reference angle. In this case, the reference angle is π / 4.
The first solution: 2x = π / 4
Dividing by 2 on both sides, we get:
x = π / 8
Additional solutions: To find additional solutions, we add integer multiples of 2π to the reference angle π / 4.
x = π / 8 + 2π * n, where n is an integer.
Step 3: Restrict the solutions to the given interval [0, 2π)
To restrict the solutions to the given interval, we need to make sure that the value of x lies between 0 and 2π.
Looking at the equation x = π / 8 + 2π * n, we can determine the range of n that satisfies this inequality:
π / 8 ≤ π / 8 + 2π * n < 2π
Simplifying the inequality, we have:
0 ≤ 2π * n < 15π / 8
Dividing both sides of the inequality by 2π, we get:
0 ≤ n < 15π / (8 * 2π)
0 ≤ n < 15 / 16
Since n is an integer, the possible values for n in the given interval are n = 0 and n = 1.
The final solutions within the interval [0, 2π) are:
x = π / 8 + 2π * 0 = π / 8
x = π / 8 + 2π * 1 = 9π / 8
Therefore, the solutions to the equation cos(2x) = (√2) / 2 on the interval [0, 2π) are x = π / 8 and x = 9π / 8.