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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Signs, monotonicity intervals, and ranges of sine and cosine

Statement

Sine is strictly increasing on each interval $[- π / 2 + 2 m π, π / 2 + 2 m π]$ and strictly decreasing on each interval $[π / 2 + 2 m π, 3 π / 2 + 2 m π]$ . Cosine is strictly decreasing on $[2 m π, (2 m + 1) π]$ and strictly increasing on $[(2 m + 1) π, (2 m + 2) π]$ . Both functions have range $[- 1, 1]$ .

Facts & Assumptions

Given: An integer $m$ .

[L1]

The zero sets, signs on the fundamental intervals, and period $2 π$ follow from The zero sets of sine and cosine and the least positive common period 2 pi.

[L2]

Quarter-turn values give the endpoint values $- 1, 0, 1$ (Quarter-turn values and shifts by pi/2 and pi).

[L3]

$\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , and the mean value theorem converts derivative sign into strict monotonicity (The derivatives of sine and cosine are cosine and minus sine, The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ).

Proof

technique · direct

1.1L1L3

On the open intervals where $\cos$ is positive respectively negative, $\sin^{'} = \cos$ makes sine strictly increasing respectively decreasing.

1.2L1L3

On the open intervals where $\sin$ is positive respectively negative, $\cos^{'} = - \sin$ makes cosine strictly decreasing respectively increasing.

2.1step 1.1step 1.2L1L2∎

The period moves these conclusions to every integer $m$ , and the endpoint values in [L2] show both ranges are exactly $[- 1, 1]$ .

Signs, monotonicity intervals, and ranges of sine and cosine

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources