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

• 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 map with two preimages but degree zero Counterexample
• A surjective map need not be a fibration Counterexample
• A twice-traversed circle has the same trace but twice the path length 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
• A closed cylinder as a finitely patched oriented surface 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
• Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
• F(x)=x² sin(1/x²) has an unbounded derivative whose Henstock–Kurzweil integral is sin 1 Example
• For every θ≥0, the unit-circle path t↦(cos t,sin t) on [0,θ] has length θ Example
• G(x)=x² sin(1/x) has a bounded derivative discontinuous at 0 that is nevertheless Riemann integrable, and Newton–Leibniz evaluates its integral Example
• Great circles as round-sphere geodesics Example
• Hopf circle fibration Example
• Mobius band as an interval bundle with monodromy Example
• Polar change of variables on a compact annular sector gives the Jacobian factor r and its area Example
• Polar coordinates are a local diffeomorphism away from zero radius Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• sin x/x has a Henstock–Kurzweil integral on [0,∞) Example
• Spherical coordinates have absolute Jacobian determinant r² sinφ away from the axis and angular seam Example
• Stokes' theorem on a flat disc and on a hemisphere with the same induced boundary circle Example
• Surface area and flux on a sphere, with scalar integrals on a hemisphere Example
• The closed ball is an elementary solid region, presented by the eight spherical octants Example
• The de Rham map on the angular form Example
• The extension of x² sin(1/x) by zero is differentiable but its derivative is discontinuous at zero Example
• The family sin(nx)sin(ny) is uniformly bounded but not equicontinuous Example
• The hyperspherical-coordinate Jacobian is the standard product of a radial power and sine powers Example
• The Mobius band presented by two regular patches, with normal comparison on the interiors of the overlap components Example

…and 41 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)