Alphabeta MathSine, Cosine, and the Definition of Pi
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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.
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✓ 12 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 12 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Lemma
The sine and cosine power series converge absolutely for every real argument
Definition
Sine and cosine defined by their real power series
Theorem
The derivatives of sine and cosine are cosine and minus sine
Theorem
Existence and uniqueness for y''=-y with prescribed initial data
Theorem
The addition formulas for sine and cosine
Corollary
Parity and the Pythagorean identity for sine and cosine
Lemma
Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3
Theorem
Cosine has a smallest positive zero, lying strictly between zero and two
Definition
Pi as twice the smallest positive zero of cosine
Theorem
Quarter-turn values and shifts by pi/2 and pi
Corollary
Pi is the first positive zero of sine
Theorem
The zero sets of sine and cosine and the least positive common period 2 pi
Theorem
Signs, monotonicity intervals, and ranges of sine and cosine
Definition
Tangent, cotangent, secant, and cosecant on their exact natural domains
Theorem
Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant
Corollary