Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-02
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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.

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sin⁡(3π/2)=−1 shows that the positive square root is not an unconditional half-angle formula

Statement refuted

The unconditional assertion sin⁡(x/2)=(1−cos⁡x)/2 for every real x is false. The conventions and prerequisite facts used below are recorded in Half-angle identities with the sign determined by the quadrant, Quarter-turn values and shifts by pi/2 and pi.

Facts & Assumptions

Given: x=3π.

Counterexample

1.1

Here sin⁡(x/2)=sin⁡(3π/2)=−1, while (1−cos⁡3π)/2=1.

given
2.1

Thus the positive square root loses the required quadrant sign.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources