"cos" 〖"α" ="1" /"2" 〗
for all 0 ≤ α ≤ 2π
This statement is true. The cosine of α is equal to 1/2 when α is equal to π/3 or 5π/3. These are the only two values of α between 0 and 2π for which the cosine is equal to 1/2. However, since the cosine function has a period of 2π, it will take on the value of 1/2 at these two points and all points that differ from them by a multiple of 2π. Therefore, cos(α) = 1/2 for all 0 ≤ α ≤ 2π.
To solve for α when cos(α) = 1/2, we need to find the angles between 0 and 2π where the cosine function equals 1/2.
The values of α that satisfy this equation are the angles whose cosine is 1/2. This means that the terminal side of each angle intersects the unit circle at a point where the x-coordinate is 1/2.
The unit circle is a circle with radius 1 centered at the origin (0, 0) in a coordinate plane. From the unit circle, we know that the cosine of an angle is equal to the x-coordinate of the point where the terminal side of the angle intersects the unit circle.
To find the angles where cos(α) = 1/2, we need to determine when the x-coordinate of these points is 1/2.
The unit circle can be divided into four quadrants, each representing one of the four trigonometric functions: sine, cosine, tangent, and cotangent. In the first quadrant, the angles have positive x and y coordinates. In the second quadrant, the angles have negative x and positive y coordinates. In the third quadrant, the angles have negative x and y coordinates. In the fourth quadrant, the angles have positive x and negative y coordinates.
Since cos(α) = 1/2, we are looking for angles where the x-coordinate of the point of intersection is 1/2. The values of α in radians that satisfy this condition are α = π/3 and α = 5π/3.
In degrees, the values of α that satisfy cos(α) = 1/2 are α = 60° and α = 300°.