Alphabeta MathHow 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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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is a bijection from onto the real unit circle
Statement
The map is a bijection from onto . The conventions and prerequisite facts used below are recorded in Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Pi as twice the smallest positive zero of cosine.
Facts & Assumptions
Given: A point on or parameters .
Proof
The Pythagorean identity puts every image point on .
The range and sign theorem supplies an angle in the stated half-open interval with prescribed cosine and the compatible sine, proving surjectivity.
If two parameters have the same pair, the sine--cosine period theorem says their difference is a multiple of ; the half-open interval forces that multiple to be zero.
Depends on
Used by
- Every a cos x+b sin x has the amplitude-phase form R cos(x-φ) Corollary
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- Every nonzero complex number has a unique polar form r(cosθ+i sinθ) with r>0 and -π<θ≤π Theorem
- ker(exp)=2π iℤ, and exp z=exp w exactly when z-w∈2π iℤ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 59 results over 23 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)