The Secant, Cosecant, and Cotangent Functions
Secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent, taken value by value. That single fact fixes their whole geometry: asymptotes stand over the parent function’s zeros (division by zero), the branches touch the parent at its peaks (the reciprocal of ±1 is ±1), and sec and csc never enter the band between −1 and 1 because their parents never leave it.
The mistakes are two. Graph errors: putting csc’s asymptotes at sine’s peaks instead of its zeros, giving sec the range of cos, or handing cot the wrong period (it repeats every π). And the notation collision: reading csc as sin⁻¹, when one flips a value and the other recovers an angle. Both are drilled head-on in the lesson.
The work
3 ways in · any order
Lesson
The Secant, Cosecant, and Cotangent Functions
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One over cosine, one over sine, cosine over sine. The lesson builds all three reciprocal functions from values you already know, parks the asymptotes over the parent zeros where they belong, maps the forbidden range band, and separates reciprocal from inverse for good. Ten scenarios close it out.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 3.11 misconceptions: asymptotes and ranges assigned to the wrong graph features, and the reciprocal-vs-inverse notation collision. 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.