Tangent and Cotangent Graphs

Pre-Calculus · Axiom Academy

LESSON Tangent and Cotangent Graphs Two curves that don't wave — they race to infinity. See where the asymptotes come from and why the period is , not . The tangent function is the ratio of sine to cosine. Wherever the denominator vanishes and is undefined — the curve shoots off to against a vertical asymptote . The animation drives a point along (purple); watch the tangent value (gold) launch toward infinity exactly as touches the axis. Asymptotes of live exactly at those zeros One full copy of the tangent lives on a single branch between consecutive asymptotes — the interval . Shift that branch right by exactly and it lands perfectly on the next branch. That is what period means: . The animation slides the highlighted branch over by so you can watch it coincide. One branch repeats every — half the period of sine and cosine. All reals except (the asymptotes). Each branch climbs from up to — strictly increasing. — symmetric through the origin; passes through (0,0) . On the central branch the curve passes through the origin and rises through , since . Every other branch is just this one shifted by a multiple of . Cotangent flips the ratio: . Now the denominator is , so the asymptotes move to where — the multiples of . And because the value runs from down to across each branch, cotangent is decreasing . The animation drives a point along (purple) and shows (gold) diving from to . Asymptotes of sit at the multiples of

This is the written version of the interactive lesson above. See the full Pre-Calculus course.