Problem

Solve \(\cos(θ)~=~1/2\) for \(θ\) in \([0,~2\pi~)\). Use the diagram to state the reference angle, valid unit-circle locations, and complete solution set.

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What this problem is really about

Treat a cosine equation as a vertical-line search on the unit circle because cosine fixes the horizontal coordinate. Find the reference angle from the magnitude, use every quadrant with the required horizontal sign, and keep only angles in the stated half-open period. The line's circle intersections verify that the solution set is complete.

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The reference angle is pi/3; the valid locations are Quadrants I and IV; the solution set is pi/3, 5pi/3.

The reference angle is pi/3; the selected locations are Quadrants II and III; the solution set is 2pi/3, 4pi/3.

The reference angle is pi/3; the selected locations are Quadrants I and IV plus the excluded endpoint; the solution set is pi/3, 5pi/3, 2pi.

The reference angle is pi/3; the only selected location is Quadrant I; the solution set is pi/3.

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Solve \(\cos(θ)~=~1/2\) for \(θ\) in \([0,~2\pi~)\). Use the diagram to state the reference angle, valid unit-circle locations, and complete solution set.

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