DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Pi as twice the smallest positive zero of cosine

Definition

Let $\gamma$ be the unique smallest positive zero of cosine supplied by Cosine has a smallest positive zero, lying strictly between zero and two. Define $\pi:=2\gamma.$

Thus $\pi>0$ and $\pi/2=\gamma$ .

Depends on

Cosine has a smallest positive zero, lying strictly between zero and two

Used by

exp(x+iy)=eˣ(cos y+i sin y), |exp(x+iy)|=eˣ, and e^iπ+1=0 Corollary
Pi is the first positive zero of sine Corollary
The Bartle-Sherbert bounds 2.828 < pi < 3.185 Example
A sector-area squeeze proves lim sin(x)/x=1 without first calibrating angle measure False statement
Quarter-turn values and shifts by pi/2 and pi Theorem
t↦(cos t,sin t) is a bijection from [0,2π) onto the real unit circle Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)