Signs, monotonicity intervals, and ranges of sine and cosine

Depends on

• The zero sets of sine and cosine and the least positive common period 2 pi
• Quarter-turn values and shifts by pi/2 and pi
• The derivatives of sine and cosine are cosine and minus sine
• The mean value theorem, as the case $g(x) = x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a < b$ and differentiable on $(a,b)$ there is $c \in (a,b)$ with $f(b) - f(a) = f'(c)(b-a)$

Used by

• Buffered certified rational phase approximation Algorithm
• Exact success amplification by attenuating the good branch Algorithm
• Szegedy-walk marked-set detection Algorithm
• A real-valued optimum is not a fractional Grover iteration Counterexample
• An invertible derivative at one point does not give a local inverse without C¹ regularity Counterexample
• Antipodal points on a round sphere have many minimizing geodesics Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• I₂(5) admits no crystallographic scaling and no reduced crystallographic root system with that base pairing Counterexample
• Sharp frequency cutoffs have kernels that are not in L1 Counterexample
• The final-time face is not part of the parabolic boundary Counterexample
• Arcsine-weighted inner product on the Chebyshev interval Definition
• Best uniform polynomial error and Chebyshev interpolation operators Definition
• Chebyshev–Gauss–Lobatto collocation grid Definition
• Principal inverse sine and inverse cosine Definition
• The angular path metric, the Euclidean cone and spherical joins Definition
• A circle of circumference ℓ<2π fails CAT(1) Example
• A cycle and an overlong arm: explicit non-positive witnesses Example
• A pure two-strand full twist as an ordered loop Example
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• A vector with mixed signs is not a root, while every root has a sign Example
• All meets and joins of the right weak order of A₂ (S₃), with the left order and the inversion sets compared Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• Analytic and nonsmooth data have geometric and algebraic Chebyshev coefficient decay Example
• B-tilde versus C-tilde: the n=2 coincidence and the duality behind the difference Example
• Bₙ and Cₙ define the same Coxeter diagram and the same Coxeter group Example
• Chamber faces and their stabilizers in A₂ Example
• Characteristics of a monomial, an exponential and a tangent Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Chebyshev–Lobatto nodes cluster quadratically at the endpoints Example
• Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
• Deficiencies of the exponential and sine Example
• Degree-one root and Lobatto interpolation norms Example
• Dihedral diagrams I₂(m): Gram determinants, the infinite case, and the low-rank coincidences Example
• Exact Grover rotation for four addresses and one marked item Example
• Exact sine and cosine values at π/10, π/5, and 2π/5 Example
• G2 from I2(6): the scaled realization and its twelve roots Example
• Gram determinants and principal minors of the non-crystallographic types H₃ and H₄ Example
• Link edge lengths versus dihedral mirror angles in type I₂(m) Example
• Midpoint iteration on a small equilateral spherical triangle contracts geometrically to its centre Example
• Normal coordinates on the round sphere Example

…and 92 more results.

Dependency tree · two levels

23 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)