CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Pi is the first positive zero of sine

Statement

$\sin π = 0$ , and $\sin x > 0$ for every $x$ with $0 < x < π$ . Thus $π$ is the first positive zero of sine.

Facts & Assumptions

Given: $π = 2 γ$ .

[L1]

The shift identities give $\sin π = 0$ and $\sin (γ + t) = \cos t$ (Quarter-turn values and shifts by pi/2 and pi).

[L2]

$\sin x > 0$ for $0 < x \leq γ$ , and $\cos t > 0$ for $0 \leq t < γ$ (Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3, Pi as twice the smallest positive zero of cosine).

Proof

technique · direct

1.1

The first assertion is in [L1].

1.2

If $0 < x \leq γ$ , positivity follows from [L2]; if $γ < x < 2 γ$ , write $x = γ + t$ with $0 < t < γ$ , and [L1] and [L2] give $\sin x = \cos t > 0$ .

L1L2cases

2.1

These cases cover $0 < x < π = 2 γ$ , proving that $π$ is the first positive sine zero.

step 1.1step 1.2cases-exhaustive∎

Depends on

Used by

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)