CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Pi is the first positive zero of sine

Statement

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

Facts & Assumptions

Given: $\pi=2\gamma$ .

[L1]

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

[L2]

$\sin x>0$ for $0<x\le\gamma$ , and $\cos t>0$ for $0\le t<\gamma$ (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 $0<x\le\gamma$ , positivity follows from [L2]; if $\gamma<x<2\gamma$ , write $x=\gamma+t$ with $0<t<\gamma$ , and [L1] and [L2] give $\sin x=\cos t>0$ .

L1L2cases

2.1

These cases cover $0<x<\pi=2\gamma$ , proving that $\pi$ is the first positive sine zero.

step 1.1step 1.2cases-exhaustive∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 13 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)