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

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

Statement

There is a unique $\gamma\in(0,2)$ with $\cos\gamma=0$ . It is the smallest positive zero of cosine.

Facts & Assumptions

Given: The cosine function.

[L1]

$\cos0=1$ , $\cos2\le-1/3$ , and $\cos$ is strictly decreasing on $[0,2]$ (Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3).

[L2]

Power-series sums are continuous on their interval of convergence (The sum of a real power series is continuous at every point strictly inside its interval of convergence).

[L3]

A continuous real function whose endpoint values bracket zero has a zero between them (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$ ).

Proof

technique · direct

1.1

Continuity, $\cos0>0$ , and $\cos2<0$ give some $\gamma\in(0,2)$ with $\cos\gamma=0$ .

L1L2L3

2.1

Strict decrease on $[0,2]$ makes this zero unique and gives $\cos x>0$ for $0\le x<\gamma$ .

step 1.1L1

3.1

No positive number below $\gamma$ is a zero, so $\gamma$ is the smallest positive zero.

step 2.1∎

Depends on

Used by

Pi as twice the smallest positive zero of cosine Definition
Quarter-turn values and shifts by pi/2 and pi Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 93 results over 17 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)