Sine and cosine defined by their real power series

Depends on

• A real power series about a centre, its interval of convergence, and its radius in $[0,+\infty]$
• The sine and cosine power series converge absolutely for every real argument

Used by

• Buffered certified rational phase approximation Algorithm
• Generator convention matters Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• DCT-II and DST-I with fixed even and odd extension conventions Definition
• Periodic Fourier collocation grid and even-grid Nyquist convention Definition
• Spherical Bessel and outgoing Hankel function conventions Definition
• The angular path metric, the Euclidean cone and spherical joins Definition
• The classical Weierstrass function Definition
• The real Coxeter form, its radical, reflections, and form-preserving maps Definition
• Type-I cosine transform for Chebyshev–Lobatto data Definition
• An indefinite Coxeter form: infinite, but not of affine type Example
• Dihedral diagrams I₂(m): Gram determinants, the infinite case, and the low-rank coincidences Example
• Rauch comparison between euclidean and spherical geodesics Example
• The A-tilde 1 infinity edge, separated from finite dihedral families and from the 4-edge Example
• The Bartle-Sherbert bounds 2.828 < pi < 3.185 Example
• The invariants and the coinvariant Hilbert series of I₂(m): an explicit computation and the noncrystallographic contrast Example
• Two independent units in a real cubic field Example
• Higher sectional curvature makes jacobi fields spread faster False statement
• Cartan-number products, allowed edge labels, tree scalings and reflection stability Lemma
• Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences Lemma
• Enumeration of the connected positive semidefinite corank-one diagrams Lemma
• Exceptional parabolic-orbit length certificates for E6, E7, E8, F4, H3 and H4 Lemma
• Exclusions for positive definite diagrams: trees, valency, labels, chains and arms Lemma
• Finite exact comparison of real cyclotomic values Lemma
• Minimum nonshrinkable loops, radial vertex cones, and the excursion of length π Lemma
• Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order Lemma
• Sine is positive and cosine is strictly decreasing on (0,2), with cos 2 at most -1/3 Lemma
• Terminating certified rounding for a counting estimate Lemma
• The Coxeter elements of the classical types Aₙ, Bₙ, Dₙ and I₂(m): characteristic polynomials, orders and spectral exponents from their reflection models Lemma
• The half-turn constant exceeds √2 Lemma
• The six exceptional Coxeter spectra: characteristic polynomials, orders and spectral exponents from exact matrices Lemma
• Correctness and arithmetic resource bound for buffered phase approximation Theorem
• Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
• Pauli formula for an axis rotation Theorem
• Relative approximate counting success guarantee Theorem
• Soundness and eventual absolute precision for spherical Bessel enclosures Theorem
• Standard Maclaurin expansions Theorem
• The cone and join metrics and the local product chart of a polyhedral gluing Theorem
• The derivatives of sine and cosine are cosine and minus sine Theorem

Dependency tree · two levels

11 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)
• C. Schmeiser, Introduction to Analysis (standard reference, not scraped)