Problem

Solve \(\sin(θ)~=~1/2\) 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 sine equation on one full period as a horizontal-line search on the unit circle: sine fixes the vertical height. Use the magnitude to identify a reference angle, then place that height in every quadrant with the required vertical sign. Finally, enforce the interval endpoints and confirm that every circle intersection has been counted.

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The reference angle is pi/6; the only selected location is Quadrant I; the solution set is pi/6.

The reference angle is pi/6; the selected locations are Quadrants III and IV; the solution set is 7pi/6, 11pi/6.

The reference angle is pi/6; the valid locations are Quadrants I and II; the solution set is pi/6, 5pi/6.

The reference angle is pi/6; the selected locations are Quadrants I and II plus the excluded endpoint; the solution set is pi/6, 5pi/6, 2pi.

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Solve \(\sin(θ)~=~1/2\) 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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