Construct secant and cosecant graphs

Try

Secant and cosecant are reciprocals: sec x=1/cos x and csc x=1/sin x. Start from exact unit-circle values instead of memorizing a collection of disconnected curves. Cosine zero means secant is undefined; cosine near zero gives a large-magnitude reciprocal. Cosine ±1 gives secant ±1, the nearest its branches come to zero. These facts determine both excluded inputs and branch vertices.

The transformed example S(x)=1−2sec x has S(0)=−1 and S(π)=S(−π)=3. Its coefficient −2 flips the reciprocal branches. The central branch, where cosine is positive, lies at or below −1 and reaches a maximum of −1 at zero. This maximum is attained; the vertical asymptotes at ±π/2 are not points on the graph.

Predict

Can S take the value 1, its vertical shift? Predict before solving. Equality would require sec x=0, but a reciprocal of a finite nonzero cosine cannot be zero. The shift line is not necessarily an attainable midline crossing, as it was for a sine graph.

For all allowed inputs, sec x lies in (−∞,−1] union [1,∞). Multiplying by −2 and adding 1 gives the disconnected range (−∞,−1] union [3,∞). The period remains 2π; adding π reverses the secant sign and generally changes the output. Full domain: every real x except π/2+kπ for integer k. The drawing shows one period, but this all-domain statement carries every missing point beyond the window.

Predict which directions occur near the central vertex. As x increases from −π/2 toward 0, cosine grows from zero toward 1, secant decreases from +∞ toward 1, and S increases from −∞ to −1. From 0 to π/2, the directions reverse. Thus it is increasing then decreasing, not a single monotone branch like tangent.

Experiment

Construct C(x)=−1+3csc(2x) from sine. Its poles solve sin(2x)=0, so x=kπ/2. Its period is π. On (0,π/2), sine is positive and reaches 1 at x=π/4, giving the branch minimum C=2. On (π/2,π), sine is negative and reaches −1 at x=3π/4, giving the branch maximum C=−4.

Mark both vertices and each excluded endpoint before connecting the curve. On the first branch the values descend from +∞ to 2 and then rise to +∞. On the second they rise from −∞ to −4 and fall to −∞. Every branch end lies outside any finite display height. The complete range is (−∞,−4] union [2,∞).

Change the coefficient 3 to −3 while retaining the shift −1 and input 2x. Reflect every output across y=−1. The first vertex becomes −4, now a maximum; the second becomes 2, now a minimum. Pole locations and period do not change. Plot at least one off-vertex value on each branch, for example at x=π/12 and x=7π/12, where sine is respectively 1/2 and −1/2. For the original C these outputs are 5 and −7; the latter is outside the displayed height but remains a valid point.

Understand

Reciprocation reverses order on any interval where the compared nonzero numbers have the same sign. Indeed 1/u−1/v=(v−u)/(uv), and uv>0. This algebra explains the branch directions without derivatives. On a positive cosine quarter-arc decreasing from 1 to zero, secant increases from 1 to +∞. On a negative quarter-arc increasing from −1 toward zero, secant decreases toward −∞. Then apply the sign of the output coefficient to obtain the transformed direction. Split at both poles and vertices; never compare across a pole as if the graph were continuous there.

For a nonzero scale a, the parent reciprocal range transforms under y=d+a z. Its two boundary values are d−∣a∣ and d+∣a∣, giving (−∞,d−∣a∣] union [d+∣a∣,∞). The endpoints are attained because sine and cosine attain ±1. A negative scale switches which input vertex supplies each boundary, even though the range formula has the same form.

For d+a sec(b(x−h)), with b≠0, solve b(x−h)=π/2+kπ for poles and b(x−h)=kπ for vertices. For cosecant, poles solve b(x−h)=kπ and vertices solve b(x−h)=π/2+kπ. The positive period is 2π/∣b∣. A negative input scale reverses traversal, while a negative output scale reverses vertical ordering. These are separate effects; solve input equations before describing a drawing.

Because a bounded screenshot cannot show an infinite branch, its clipped endpoint must not be reported as a maximum or minimum. Only an attained mathematical value with the stated branch comparison qualifies. A physical use may also restrict the domain further, in which case finite interval endpoints need their own consideration.

Apply

Solve S(x)≥3 on −π≤x≤π using S=1−2/cos x. Fix the cosine sign before multiplying or taking a reciprocal. Then construct a fresh graph R(x)=2+sec(x−π/4), labeling a full cycle’s poles, vertices and range.

Check the inequality and fresh construction

The inequality is sec x≤−1. It holds exactly where cosine is negative, since every negative secant is at most −1. In the stated window the result is [−π,−π/2) union (π/2,π]. Both finite window endpoints satisfy equality, while the poles remain excluded. Multiplying by cosine without deciding its sign would risk reversing the wrong inequality.

For R, poles are x=3π/4+kπ. Vertices are at x=π/4+kπ, with alternating heights 3 and 1. A full cycle from −π/4 to 7π/4 contains the upward vertex (π/4,3), downward vertex (5π/4,1), and poles at −π/4, 3π/4 and 7π/4; both window endpoints are excluded. Range is (−∞,1] union [3,∞). These two adjacent complete branches occupy the open intervals between the listed poles.

Explain the difference between a complete family rule and one displayed cycle. A full rule lists all integer-indexed exclusions; a cycle sketch illustrates them locally. Both are needed to avoid losing branches outside a convenient window.

Continue

Submit a construction with exact poles and vertices, a disconnected range, arrowed branch continuation, direction intervals and a statement of the horizontal window. Include a reciprocal sign argument for at least one branch instead of relying only on a remembered picture.

Return to tangent and cotangent branches to contrast their all-real branch ranges, or to periodic fitting when a smooth periodic formula is a candidate for noisy observations. An exact trigonometric graph and empirical evidence supporting a real model are different kinds of work.