Problem

Solve \(\tan(θ)~=~1\) 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

A tangent equation fixes the ratio of vertical to horizontal coordinates. Find the reference angle from the ratio, use the quadrants where that quotient has the required sign, and exploit tangent's half-turn period to locate the companion angle. Check that cosine is nonzero and that every result lies in the required interval.

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The reference angle is pi/4; the selected locations are Quadrants II and IV; the solution set is 3pi/4, 7pi/4.

The reference angle is pi/4; the selected locations are Quadrants I and III plus the excluded endpoint; the solution set is pi/4, 5pi/4, 2pi.

The reference angle is pi/4; the only selected location is Quadrant I; the solution set is pi/4.

The reference angle is pi/4; the valid locations are Quadrants I and III; the solution set is pi/4, 5pi/4.

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Solve \(\tan(θ)~=~1\) 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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