The addition formulas for sine and cosine

Depends on

• Existence and uniqueness for y''=-y with prescribed initial data
• The derivatives of sine and cosine are cosine and minus sine
• The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$

Used by

• Gray-Path Synthesis of a Two-Level Gate Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• Every a cos x+b sin x has the amplitude-phase form R cos(x-φ) Corollary
• Parity and the Pythagorean identity for sine and cosine Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• A real-valued optimum is not a fractional Grover iteration Counterexample
• Elliptic eigenvalues need not be simple Counterexample
• f(x+iy)=eˣ(cos 2y+i sin 2y) is continuous, satisfies f(z+w)=f(z)f(w) and f(1)=e, but is not the standard complex exponential Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
• The Coxeter nerve and its Moussong metric Definition
• A circle of circumference ℓ<2π fails CAT(1) Example
• A quadratic Fourier product aliases two high modes on an even grid Example
• A vector with mixed signs is not a root, while every root has a sign Example
• An atomic sinusoid and an AR(1) process have different spectra Example
• An indefinite Coxeter form: infinite, but not of affine type Example
• Analytic and nonsmooth data have geometric and algebraic Chebyshev coefficient decay Example
• Chamber faces and their stabilizers in A₂ Example
• Chebyshev–Lobatto nodes cluster quadratically at the endpoints Example
• Dirichlet Laplacian eigenpairs on an interval Example
• Exact Grover rotation for four addresses and one marked item Example
• L² forcing defines an H⁻¹ functional 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
• Null-homotopy versus shrinkability through short loops on S² and on a short circle Example
• Roots, inversions and chamber images in I₂(5), A₂ and infinite dihedral type Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers Example
• The finite dihedral rotation and the infinite unipotent rank-two product Example
• The invariants and the coinvariant Hilbert series of I₂(m): an explicit computation and the noncrystallographic contrast Example
• The noncrossing interval of a dihedral group: a five-reflection claw for I2(5) and its complement Example
• Toponogov comparison on a round sphere Example
• Alexandrov comparison: straightening a hinge, gluing comparison triangles, and patchwork Lemma
• Amplitude-amplification invariant plane Lemma
• Average Grover success for a uniform integer iteration count Lemma
• Canonical Banach complexification of a real Banach space Lemma
• Chebyshev bounds and endpoint derivatives used by stabilized Runge–Kutta methods Lemma
• Chebyshev polynomials are orthogonal for the arcsine weight Lemma
• Clifford+T Is Dense Projectively on One Qubit Lemma
• Comparison triangles in the Euclidean plane and the round sphere, model spaces, and CAT(0) and CAT(1) consequences Lemma
• Constant BBHT stage success beyond the threshold scale Lemma

…and 62 more results.

Dependency tree · two levels

15 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)