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.
- 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.
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
- The one-dimensional torus and its normalized Haar integral Definition
- A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
- Great circles as round-sphere geodesics Example
- Polar coordinates are a local diffeomorphism away from zero radius Example
- Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
- SU(2) to SO(3) as a covering homomorphism Example
- The closed ball is an elementary solid region, presented by the eight spherical octants Example
- The outward flux of the inverse-square field through a sphere centred at the origin is 4π Example
- Finite tori are compact Hausdorff spaces separated by characters Lemma
- Fourier uniqueness for continuous functions on the Euclidean torus Lemma
- Tangent is a continuous strictly increasing bijection from (-π/2,π/2) onto ℝ Lemma
- The plane Gaussian integral equals π by polar coordinates Lemma
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- [t]↦(cos 2π t,sin 2π t) is a homeomorphism from ℝ/ℤ to the unit circle Theorem
- A circle traversed k times has winding number k inside and 0 outside Theorem
- 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 · two levels
17 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
- NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)