Pi is the first positive zero of sine

Depends on

• Quarter-turn values and shifts by pi/2 and pi
• Pi as twice the smallest positive zero of cosine
• Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3

Used by

• A fixed Grover iteration count can fail when a is unknown Counterexample
• A map with two preimages but degree zero Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• Comparison angles of hinges, model triangle angles, and the Alexandrov upper angle Definition
• Short loops, the uniform-plus-length topology, short-loop homotopies, and nonshrinkability Definition
• The angular path metric, the Euclidean cone and spherical joins Definition
• Uniform local radii, cyclic small-mesh polygons, mesh, length, energy, the midpoint operation and the zero-limit basin Definition
• A circle of circumference ℓ<2π fails CAT(1) Example
• A complete locally CAT(0) circle whose fundamental group prevents global CAT(0) Example
• A half-circle configuration loop traces an elementary half twist Example
• A prepared state with success amplitude one half Example
• A pure two-strand full twist as an ordered loop Example
• Addition and half-angle identities compute the sine, cosine, and tangent of π/12 Example
• Equally spaced points on a metric circle: stationary energy and the equality case Example
• Index form in constant curvature Example
• Link edge lengths versus dihedral mirror angles in type I₂(m) Example
• Midpoint iteration on a small equilateral spherical triangle contracts geometrically to its centre Example
• Normal coordinates on the round sphere Example
• Null-homotopy versus shrinkability through short loops on S² and on a short circle Example
• Point pushing one puncture around another Example
• Quadratic Gauss sum at p=3 Example
• Quadratic Gauss sum at p=5 Example
• Spectral distribution and phase resolution for a two-level matrix Example
• The noncrossing interval of a dihedral group: a five-reflection claw for I2(5) and its complement Example
• The Rayleigh quotient on an interval Example
• The solid generated by rotating y=sin x on [0,π] has volume π²/2 Example
• The surface generated by rotating y=sin x on [0,π] has area 2π(√2+arsinh1) Example
• The unit circle is CAT(1) at the strict perimeter boundary Example
• The zero-length boundary: constant tuples, collapsed edges and the degeneracy of the energy decrement at L=0 Example
• Two initial intervals converge to different minima of a periodic objective Example
• Alexandrov comparison: straightening a hinge, gluing comparison triangles, and patchwork Lemma
• Amplitude-amplification invariant plane Lemma
• Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences Lemma
• Continuous Markov derivative bound on an interval Lemma
• Fresnel whole-line moment from the damped complex Gaussian Lemma
• Local CAT(1) of the l² product from a model S²× S² sine-comparison calculation Lemma
• Perturbation by a Euclidean regular polygon: comparison-disk bounds for degenerate comparison triangles Lemma
• Polygon transfer, the basin as the shrinkable class, and the short-loop criterion Lemma
• Products of CAT(0) spaces, joins of CAT(1) spaces, and round spheres Lemma
• The sharp Dirichlet Poincare inequality on an interval Lemma

…and 11 more results.

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
• C. Schmeiser, Introduction to Analysis (standard reference, not scraped)