TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Pi is equivalently the first sine zero, twice the first cosine zero, and half the least common period

Statement

For a positive real $p$ , the following are equivalent:

$p = π$ ;
$p$ is the least positive zero of sine;
$p / 2$ is the least positive zero of cosine;
$2 p$ is the least positive common period of sine and cosine.

Facts & Assumptions

Given: A positive real $p$ .

[L1]

If $γ$ is the unique least positive zero of cosine, then $π = 2 γ$ (Pi as twice the smallest positive zero of cosine).

[L2]

$\sin π = 0$ , and $\sin x > 0$ for every $0 < x < π$ ; thus $π$ is the first positive zero of sine (Pi is the first positive zero of sine).

[L3]

Both sine and cosine have period $2 π$ , and no smaller positive number is a common period (The zero sets of sine and cosine and the least positive common period 2 pi).

Proof

technique · direct

1.1

If $p = π$ , then [L2] says that $p$ is the least positive zero of sine.

givenL2

1.2

If $p = π$ , write $π = 2 γ$ as in [L1]. Then $p / 2 = γ$ , the least positive zero of cosine.

givenL1algebra

1.3

If $p = π$ , then $2 p = 2 π$ , which is the least positive common period by [L3].

givenL3algebra

1.4

Conversely, if $p$ is the least positive zero of sine, then $p = π$ because [L2] identifies $π$ as that least positive zero.

givenL2

1.5

If $p / 2$ is the least positive zero of cosine, then [L1] gives $p / 2 = π / 2$ , hence $p = π$ .

givenL1algebra

1.6

If $2 p$ is the least positive common period, then [L3] gives $2 p = 2 π$ , hence $p = π$ .

givenL3algebra

2.1

Steps 1.1 to 1.6 prove every implication to and from $p = π$ , so the four conditions are equivalent.

step 1.1step 1.2step 1.3step 1.4step 1.5step 1.6∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 15 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

J. Lebl, Basic Analysis II, section 11.4.2 (standard reference, not scraped)