Final answer:
The equation 2 cos 2θ − 1 = 0 has 2 roots within the interval 0 ≤ θ ≤ 2π, since cos 2θ = 1/2 corresponds to θ values of π/6 and 5π/6.
Step-by-step explanation:
The equation given is 2 cos 2θ − 1 = 0. To find the number of roots within the interval 0 ≤ θ ≤ 2π, first we isolate cos 2θ by adding 1 to both sides and then dividing by 2, giving us cos 2θ = 1/2. We can then use the cosine function to determine the angles that satisfy this condition within the given interval.
For cos 2θ = 1/2, we know the angle 2θ can be π/3, 5π/3 within a full circle (since cosine is positive in the first and fourth quadrants). To find θ, we divide these angles by 2: θ can be π/6 or 5π/6. Since the interval is 0 to 2π, we will have these solutions each occur once in the interval, which implies we have 2 roots for the given equation within the specified interval.