Tangent and Its Asymptotes

Algebra 2 · Axiom Academy

LESSON Tangent and Its Asymptotes Define tangent as sine over cosine, see why its graph races off to infinity wherever cosine vanishes, then read its shape between the asymptotes. 1. Tangent Is the Slope of the Terminal Ray Put an angle in standard position on the unit circle. Its terminal ray meets the circle at the point — so the ray's rise is and its run is . The slope of that ray, rise over run, is exactly . the ratio of the coordinates on the unit circle …which is the slope of the terminal ray Tangent is a ratio with in the denominator. Wherever , the division is undefined and the graph shoots off to infinity. On the unit circle at (the top) and (the bottom) — that is, at . Plotting against gives a chain of identical branches. Each branch climbs from to , crosses zero at a multiple of , and is walled off by vertical asymptotes at . Because every branch copies the last, tangent has period — it repeats every , not like sine and cosine. Repeats every — half of sine's and cosine's . On every branch rises from up to . On the branch running from 0 up to the asymptote at , tangent climbs smoothly through a handful of exact special-angle values before blowing up. You've seen tangent as a slope, watched its asymptotes appear where cosine vanishes, and mapped its repeating graph and key values. Scroll up to revisit any step.

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