Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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⁡xa\cos x+b\sin x has the amplitude-phase form Rcos⁡(x−ϕ)R\cos(x-\phi)

Statement

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

Facts & Assumptions

Given: Reals a,b,xa,b,x.

Proof

technique · direct
1.1

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

algebra
1.2

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

given
2.1

Expand Rcos⁡(x−ϕ)R\cos(x-\phi) and substitute the two defining coordinates of ϕ\phi.

algebra∎

Depends on

Used by

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Sources