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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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

1.1L1

The first assertion is in [L1].

1.2L1L2cases

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 $\sin x = \cos t > 0$ .

2.1step 1.1step 1.2cases-exhaustive∎

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

Pi is the first positive zero of sine

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources