Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck 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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Every acos⁡x+bsin⁡x has the amplitude-phase form Rcos⁡(x−ϕ)

Statement

For reals a,b not both zero, put R=a2+b2>0. There is a unique ϕ∈[0,2π) with cos⁡ϕ=a/R and sin⁡ϕ=b/R, and acos⁡x+bsin⁡x=Rcos⁡(x−ϕ) for every real x. For a=b=0 the left side is identically zero. The conventions and prerequisite facts used below are recorded in t↦(cos⁡t,sin⁡t) is a bijection from [0,2π) onto the real unit circle, Square roots exist: a unique a≥0 with (a)2=a; the positives are {x2:x≠0}, The addition formulas for sine and cosine.

Facts & Assumptions

Given: Reals a,b,x.

Proof

technique · direct
1.1

If a=b=0, the claimed zero identity is immediate.

algebra
1.2

Otherwise R>0 and (a/R)2+(b/R)2=1, so the unit-circle parametrization gives the stated unique ϕ.

given
2.1

Expand Rcos⁡(x−ϕ) and substitute the two defining coordinates of ϕ.

algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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