Parity and the Pythagorean identity for sine and cosine

Depends on

• The addition formulas for sine and cosine
• The derivatives of sine and cosine are cosine and minus sine
• Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$
• A function differentiable at $c$ is continuous at $c$
• A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant

Used by

• Exact success amplification by attenuating the good branch Algorithm
• Known-amplitude amplification with integer rounding Algorithm
• Every continuous function on [0,1] is uniformly approximated by everywhere-differentiable functions whose derivative vanishes at a prescribed point Corollary
• Sine and cosine are 1-Lipschitz on ℝ Corollary
• The normal component of the curl is the limiting circulation per unit area of shrinking discs Corollary
• x sin(1/x) extended by zero is continuous but not differentiable at zero Corollary
• A curl-free C¹ field on the complement of a line that is not conservative Counterexample
• A fixed Grover iteration count can fail when a is unknown Counterexample
• A map with two preimages but degree zero Counterexample
• A radial Poisson limit does not control a tangential path Counterexample
• A surjective map need not be a fibration Counterexample
• A twice-traversed circle has the same trace but twice the path length Counterexample
• Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
• Flat hyperplanes do not have spherical stationary-phase decay Counterexample
• Hadamard instability despite analytic solvability Counterexample
• Polar coordinates on a full closed angular period are not injective and are singular at radius zero Counterexample
• Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
• The circular curve defeats the equality form of the vector-valued mean value theorem Counterexample
• The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable Counterexample
• The vortex field is closed but not exact on the punctured plane Counterexample
• Volterra's function is differentiable everywhere with bounded derivative, but its derivative is not Riemann integrable Counterexample
• Principal inverse sine and inverse cosine Definition
• Radian angle by unit-circle arc length Definition
• The classical Weierstrass function Definition
• The normalised sinc function Definition
• A closed cylinder as a finitely patched oriented surface Example
• A cycle and an overlong arm: explicit non-positive witnesses Example
• A moved-space intersection in A₃ that is not the meet Example
• A prepared state with success amplitude one half Example
• A right circular cylinder is an elementary solid region, presented by two caps and four side quarters Example
• A vector line integral around the vortex counts repeated traversals 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
• Analytic and nonsmooth data have geometric and algebraic Chebyshev coefficient decay Example
• Bₙ and Cₙ define the same Coxeter diagram and the same Coxeter group Example
• Chamber faces and their stabilizers in A₂ Example
• Chebyshev extremal nodes converge to the arcsine equilibrium measure Example
• Cut locus of a point on a round sphere Example
• Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
• Dihedral diagrams I₂(m): Gram determinants, the infinite case, and the low-rank coincidences Example

…and 141 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)