The Tangent Function
Tangent is the slope of the unit-circle ray: sine over cosine. The quotient structure gives it everything that separates it from the waves. A half turn flips both coordinates, so the slope repeats every pi, half of sine’s period. Cosine in the denominator plants vertical asymptotes at odd multiples of pi over 2, the numerator puts zeros at multiples of pi, and each branch climbs through every real value, always increasing, with no amplitude at all.
The mistakes here are imported sine habits: quoting period 2pi, granting a tangent graph an amplitude or a bounded range, placing the asymptotes where sine dies instead of cosine, and distributing tangent over sums or doubled angles. The lesson names each import and drills the quotient picture that replaces it.
The work
3 ways in · any order
Lesson
The Tangent Function
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Sine over cosine makes tangent a slope, and the slope picture runs the whole topic: period pi, asymptotes where cosine vanishes, zeros where sine does, branches that climb through every real value. Ten scenarios drill the behavior facts that sine habits get wrong.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 3.8 misconceptions: sine-shaped expectations about tangent's period, amplitude, and asymptotes, and illegal linearity moves like splitting tan of a sum. Results route you to the drills that fix what fired.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.