Secant and Cosecant Graphs

Pre-Calculus · Axiom Academy

LESSON Secant and Cosecant Graphs Build the reciprocal curves straight from sine and cosine — U-shaped branches that hug the asymptotes and touch the parent at every peak and valley. Start with the sine wave, then take its reciprocal point by point. Where dips to 0 — at every multiple of — dividing by it sends off to a vertical asymptote . Between those zeros the reciprocal forms a U-shaped branch sitting inside each hump of the sine, just touching it where . Parent: the sine wave (height between -1 and 1 ) Reciprocal: undefined at the sine's zeros Secant works the same way, but on cosine. Because at the odd multiples of , the asymptotes shift over to . Each U-branch nestles in a hump of cosine and touches it where — at x = 0 (touching +1 ) and (touching -1 ). The U bottoms out on cosine where : (0,1) and . — the graph is a mirror image across the y -axis. Identical U-pattern to cosecant, just slid over by . At x = 0 , , so — the U sits right on the cosine peak. Step a little either way and shrinks below 1 , so grows above 1 : the branch curves upward, away from the axis. 3. Why the Branches Run to Infinity Watch a single input slide across one hump. The parent height shrinks toward 0 near the asymptote and grows toward 1 at the peak — and the reciprocal does the exact opposite. Dividing by a tiny number gives a huge result , so as the parent is squeezed to 0 the reciprocal is launched to infinity; when the parent is largest at , the reciprocal is smallest at too.

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