The derivatives of sine and cosine are cosine and minus sine

Depends on

• Sine and cosine defined by their real power series
• The sine and cosine power series converge absolutely for every real argument
• Inside its radius a real power series may be differentiated term by term, and the differentiated series has the same radius

Used by

• A uniform limit of smooth functions need not be differentiable anywhere Corollary
• Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin Corollary
• Every continuous function on [0,1] is uniformly approximated by everywhere-differentiable functions whose derivative vanishes at a prescribed point Corollary
• Parity and the Pythagorean identity for sine and cosine Corollary
• Sine and cosine are 1-Lipschitz on ℝ Corollary
• The limit of sin x divided by x at zero is one 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 figure-eight curve is an immersed image but not an embedded submanifold Counterexample
• A map with two preimages but degree zero Counterexample
• A twice-traversed circle has the same trace but twice the path length Counterexample
• An invertible derivative at one point does not give a local inverse without C¹ regularity 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
• Hadamard instability despite analytic solvability Counterexample
• Infinite variance can defeat square-root-n CLT scaling Counterexample
• Irrational rotation is ergodic but not weakly mixing Counterexample
• Pointwise limit discontinuous at zero signals mass escape Counterexample
• Principal arcsine has no finite derivative at -1 or 1 Counterexample
• The circular curve defeats the equality form of the vector-valued mean value theorem 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 one-dimensional torus and its normalized Haar integral Definition
• A closed cylinder as a finitely patched oriented surface Example
• A continuous argument computed along a spiralling contour 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
• Cauchy law and its characteristic function Example
• Characteristic function of a gaussian law Example
• Characteristic function of the uniform law Example
• Cylindrical coordinates have absolute Jacobian determinant r on an injective compact box Example
• Density inversion for a triangular characteristic function 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
• Independent sums via characteristic functions Example
• Machin's formula π/4=4 arctan(1/5)-arctan(1/239) Example

…and 81 more results.

Dependency tree · two levels

13 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)