cos a = 1/2 for all 0 ≤ α ≤ 2π
This statement is false.
At α = 0 and α = 2π, cos α = 1, but for other values of α, cos α can take on different values.
I think what they meant was
cos a = 1/2 for 0 ≤ a ≤ 2π
That is, all solutions in the interval [0,2π)
That would be a = π/3 or 5π/3
Yes, that statement is true. The solutions for cos a = 1/2 in the interval [0, 2π) are indeed a = π/3 or 5π/3.
To find the values of α where cos α = 1/2 for 0 ≤ α ≤ 2π, follow these steps:
Step 1: Recall the special values of cos α for angles in the interval [0, 2π]. The special values of cos α for α in this interval are 1, 1/2, 0, -1/2, and -1.
Step 2: Since cos a = 1/2, we need to determine the angles where cos α = 1/2.
Step 3: From step 1, the special value 1/2 corresponds to the angle π/3 in the interval [0, 2π].
Step 4: Add π to the angle π/3 to find another angle where cos α = 1/2. Thus, π/3 + π = 4π/3.
Step 5: Since the interval for α is [0, 2π], these are the only two angles within that interval where cos α = 1/2.
Therefore, for 0 ≤ α ≤ 2π, the values of α where cos α = 1/2 are π/3 and 4π/3.