Solving cos(x) ≤ 1/2

Trigonometry · Axiom Academy

Finding all angles where the cosine is less than or equal to 1/2 by identifying boundary points and testing regions on the unit circle. On the unit circle, cosine represents the x-coordinate of a point. We need to find all angles where this x-coordinate is at most 1/2 (including 1/2 itself). 2. Finding the Boundary Points First, we solve the equation cos(x) = 1/2 to find where the boundary occurs. These are the angles where the point on the unit circle has exactly x = 1/2. x = 5π/3 (300°) in Quadrant IV The boundary points divide the circle into two regions. We test a point in each region to determine which satisfies cos(x) ≤ 1/2. The solution in the interval [0, 2π) is the arc from π/3 to 5π/3, moving counterclockwise through the "left side" of the circle. 5. General Solution (All Real Numbers) Since cosine has period 2π, we can extend this solution to all real numbers by adding integer multiples of 2π. x ∈ [π/3 + 2πn, 5π/3 + 2πn] where n ∈ ℤ

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