46) Determine solutions for each equation in the interval 0 ≤ x ≤ 2π, to the nearest hundredth of a radian.
Give exact answers where possible.
i) 2 cos(2x) = 1
To solve the equation 2 cos(2x) = 1, divide both sides by 2:
cos(2x) = 1/2
To find the solutions in the specified interval, we need to find the angles whose cosine is 1/2. Since the cosine function has a period of 2π, the solutions will be found in the interval 0 ≤ x ≤ 2π.
The values for which cos(2x) = 1/2 are π/3 and 5π/3.
So the solutions in the interval 0 ≤ x ≤ 2π are:
x = π/3 and x = 5π/3.